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The minimum maximal-cube covering approach to switching The minimum maximal-cube covering approach to switching

theory theory

Yu-tsang Gene Hwang New Jersey Institute of Technology

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H w a n g , Y u -t s a n g G e n e

THE MINIMUM MAXIMAL-CUBE COVERING APPROACH TO SWITCHING THEORY

New Jersey Institute o f Technology D.ENG.SC. 1980

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THE MINIMUM MAXIMAL-CUBE COVERING APPROACH TO SWITCHING THEORY

BYYU-TSANG HWANG

A DISSERTATION PRESENTED IN PARTIAL FULFILLMENT OF

THE REQUIREMENTS FOR THE DEGREEOF

DOCTOR OF SCIENCE IN ELECTRICAL ENGINEERINGAT

NEW JERSEY INSTITUTE OF TECHNOLOGY

This dissertation is to be used only with due regard to the rights of the author. Bibliographical references may be noted, but passages must not be copied without permission of the College and without credit being given in subsequent written or published word.

Newark, New Jersey1980

APPROVAL OF DISSERTATION

THE MINIMUM MAXIMAL-CUBE COVERING

APPROACH TO SWITCHING THEORY

BY

YU-TSANG HWANG

FOR

DEPARTMENT OF ELECTRICAL ENGINEERING

NEW JERSEY INSTITUTE OF TECHNOLOGY

BY

FACULTY COMMITTEE

APPROVED: CHAIRMAN

NEWARK, NEW JERSEY

MAY, 1980

ABSTRACT

This dissertation is devoted to the construction of a new algebraic structure for the K-map so that algorithms and theorems can be derived for the minimization of Boolean functions and at the same time provide for the manipulation of Boolean functions as freely as one could on the K-map.

Computer programs have been included to demonstrate the theory and numerous examples of various complexity have been worked out in the appendices.

iv

PREFACE

With the advent of microprocessors and programmable logical arrays, low speed logic circuits can now be real­ized without performing the tedious and complicated pro­cess associated with switching-function minimization. However, as speeds and complexities of switching functions become ever faster and more complex, the classical two- level random logic approach, far from being obsolete, deserves serious consideration. Having the least prop­agation delay is sufficient advantage for the two-level AND/OR gating logic to be considered in future switching circuits. In an ultra-fast system, if propagation delays are of the same order of magnitude as pulse widths, as the clock period approaches twice the smallest pulse width, the system cycle time can actually become faster than the execution time of a microprocessor or the access time of a logic array. When this happens, two-level ran­dom logic is the only alternative. In the report of a study conducted by Rudolf E. Thun and Edward T. Lewis1 atRaytheon Company, the impact of gigahertz logic on system

2design is discussed. Berthod G. Bosch also did a study on the prospect of ultra-fast logic systems in the 1980's. These studies indicate that the coming of ultra-fast logic systems in the 1980's is clear and certain.

V

Research ranging from the hardware packaging to the new scope of the applications is in full swing; the Military VLSI ultra-fast speed logic program is also progressing swiftly.

The most recent attempts in the classical two-level logic minimizations are to unify the two steps of Quine's original concepts, i.e. to search the minimum set directly without finding the prime implicant tables. McKinney's3

attempt with the direct search method, Curtis '*4 with the adjacent table method and Arevalo's5 four-tuple method are all in this trend. The most natural and logical route is actually the algebraic equivalent to the Karnaugh-map method. However, the lack of an algebraic structure prevents one from manipulating the K-map analytically.

This dissertation is devoted to the construction of a new algebraic structure for the K-map so that algorithms and theorems can be derived for the minimization of Boolean functions and at the same time provide for the manipulation of Boolean functions as freely as one could on the K-map. We call it the minimum maximal-cube covering approach. 6

In Chapter One, a brief historical survey and review of switching circuits, Boolean algebra and Boolean function minimization is presented. The comparison of

vi

various approaches to the minimization of switching circuits leads to a set of most desirable characteristics of the mechanics of minimization of switching circuits.

Chapter Two is devoted to the construction of the algebraic structure and the theoretical foundation of the minimum maximal-cube covering approach to switching theory. Starting from definitions and theorems of minimum maximal- cube covering approach, a deeper and richer insight of switching theory is shown. Applications and examples will be demonstrated in Chapters Three, Four and Five.

Chapter Three covers the algorithms of a cube graph approach with demonstrated example problems.

Chapter Four covers a different and more computer- oriented algorithm, the algebraic-geometric approach. Examples, problems and demonstrative computer programs are also shown in Chapter Four.

In Chapter Five, applications in other areas are explored - the decomposition of Boolean functions, the symmetry of Boolean functions, the sequential circuits, the coding theory, et cetera.

In Chapter Six, the concluding chapter, we summarize the results and also look beyond this dessertation to the prospective of minimum maximal cube covering approach.

vii

1 Thun, E. Rudolf and Edward T. Lewis, "Advanced Packag­ing Techniques, Final Report on Task I" AFAL-TR-77-165, part I, Raytheon Company, Missile Systems Division, Hartwell Road, Bedford, Massachusetts, 01730 pp. 10-16, August 1977.2 Bosch, B. G., "Gigabit Electronics", Proc. 1977 Europ. Microwave Conference, Copenhagen. (invited paper) pp.1-6.3 McKinney, M. H., "A directed search algorithm for the canonical minimization of switching functions", Ph.D. dissertation, Texas A&M University, College Station,TX, 1974. Curtis, H. A., "Adjacency table Method of Deriving Minimal Sums," IEEE Transaction on Computers Vol. C-26,No. 11, Nov. 1977, pp. 1136-1141.5 Arevalo, Z and Bredeson, J. G. "A Method to Simplify a Boolean Function into a near minimal sum-of-products for programmable logic arrays", IEEE Transaction of Computers Vol C-27, No. 11, Nov. 1978, pp. 1026-1039.0 For detailed explanation of the meaning of Minimum Maximal-Cube Covering, please refer to Chapter 1 and Chapter 2.

viii

DEDICATION

To my wife, Rene, for her patience, her understanding, and her encouragement.

ACKNOWLEDGEMENTS

The author gratefully acknowledges the guidance and assis­tance of Professor Robert E. Anderson. He also expresses his gratitude to Drs. Kenneth Sohn, William J. Troop, and Simon Cohen.

The author also acknowledges the finacial support of New Jersey Institute of Technology and the technical support of Aeroflex Laboratories,Inc.

TABLE OF CONTENTSPagd

A B S T R A C T .................................................. 111p r e f a c e .............................................. 1VD E D I C A T I O N .............................................. V111

*i vACKNOWLEGEMENT .........................................LIST OF F I G U R E S ......................................... xiiLIST OF T A B L E S ......................................... xivCHAPTER

I HISTORICAL REVIEW OF SWITCHING THEORY ANDMINIMIZATION ..................................... 11.1 Historical Review1.2 Comparison of Various Approaches1.3 Desirable Characteristic and Discussion

II ALGEBRAIC STRUCTURE AND THEORY.................. 192.1 The Minimum Maximal-Cube Covering2.2 Algebraic Structure of Maximal-Cube Covering

2•3 Cube Graphs and Branching GRaphs 2.4. Geometric Structure of Maximal-Cube Covering

III THE APPLICATION OF BRANCHING GRAPHS AND CUBEG R A P H S ............................................. 6 6

3.1 Algorith for Two-level Logic Minimization3.2 Demonstrative Examples of Two-level Logics.3.3 Conclusion and Discussion

IV THE APPLICATION OF GEOMETRIC APPROACH • • • • 1 0 7

4.1 Algorithm for Two-level Logic Minimization4.2 Demonstrative examples4.3 Conclusion and Discussion

V EXTENSION OF THE MINIMUM MAXIMAL-CUBE THEORY TO HIGHER CUBES AND OTHER KIND OF SWITCHING F U C T I O N S ..........................................1 3 8

5.1 Extension to Higher cubes5.2 Extension to other kind of Switching Functions5.3 Discussion

VI SUMMARY, CONCLUSIONS AND RECOMMENDATIONS . . . 1 4 1

REFERENCE ............................................... 1 4 4

APPENDICESA. COMPUTER PROGRAMS .............................. A1B. EXAMPLE OF 4 V A R I A B L E S ........................ A17C. EXAMPLE OF 6 VARIABLES, 18 M I N T E R M S ............ A43D. EXAMPLE OF 5 VARIABLES, 20 M I N T E R M S ............ A50E. EXAMPLE OF 7 VARIABLES, 5 3 M I N T E R M S ............ A56F. EXAMPLE OF 8 VARIABLES, 126 M I N T E R M S ............ A 6 6

G. EXAMPLE OF 7 VARIABLES, 92 M I N T E R M S ............ All

G. EXAMPLE OF 9 VARIABLES, 120 ON-SET MINTERMS, AND40 DON'T CARE M I N T E R M S ....................... A95

I. EXAMPLE OF 9 VARIABLES, 5 ON-SET MINTERMS, AND503 DON'T M I N T E R M S ............................ A13 3

xii

LIST OF FIGURES

Figure 2.1

Figure 2.2

Figure 2.3 Figure 2.4

Figure 2.5

Figure 3.1 Figure 3.2

Figure 3.3 Figure 3.4

Figure 3.5 Figure 3 . 6

Figure 3.7 Figure 3.8

Figure 3.9 Figure 3.10 Figure 3.11 Figure 3.12 Figure 3.13 Figure 3.14

Matrix A, Distance Matrix for Example 2.2.1Matrix B, Distance Matrix for Example 2.2.1Karnaugh Map for Example 2.2.1The Analogy of the Awayness of a Point to a Cube in Plane GeometryThe Analogy of the Awayness of a Cube to a Cube in Plane GeometryDistance Matrix A of Example 1Distance Frequency Matrix B of Example 1Matrix L of Example 1 (3)

Table for Finding 2-Cubes Covering Point 0 in Example 1Matrix L]_ of Example 1Table for Finding 2-Cubes Covering Point 3 in Example 1Matrix L2 of Example 1Table for Finding 1-Cubes Covering Point 13 in Example 1Matrix L 3 of Example 1Karnaugh Map for Example 1Matrix A of Example 2Matrix B of Example 2Matrix L of Example 2Table for Finding 1-Cubes Covering Point 0 in Example 2

Page No,

28

29

30 58

58

697071

72

7374

75767778

80818283

X I 11

LIST OF FIGURES

Figure 3.15 Figure 3.16

Figure 3.17

Figure 3.18 Figure 3.19

Figure 3.20 Figure 3.21 Figure 3.22

Figure 3.2 3

Figure 3.24

Figure 3.25 Figure 3.26

Figure 3.27 Figure 3.2 8

Figure 3.2 9

Matrix of Example 2Table for Finding 1-Cubes Covering Point 2 of Example 2Table for Finding 1-Cubes Covering Point 4 of Example 2Matrix L 2 of Example 2Table for Finding 1-Cubes Covering Points 3 and 5 of Example 2Matrix L 3 of Example 2Karnaugh Map of Example 2Branching Graph and Cube Graph of Example 2Branching Graphs and Karnaugh Map for Example 3Branching Graphs and Karnaugh Map for Example 4Branching Graph for Example 5Branching Graph (Jointed and Covered) for Example 5Karnaugh Map for Example 5Branching Graph for Example 5 (Different but Isomorphic to Figure 25)Diagrams for Example 6(a) Matrix A.(b) Matrix B(c) Matrix L(d) Table for Finding 3-Cubes(e) Matrix(f) Table for Finding 2-Cubes(g) Matrix L2(h) Karnaugh Map

Page No.

8485

86

8788

899091

93

96

9798

99100

101— 103

LIST OF FIGURES xiv

Figure 4.1 Figure 4.2

Figure 4.3

Page No.Flow Chart of Main Program 112Flow Chart of Subroutine Search for All Essential Core Maximal Cubes (called by main program in Figure 1, 113Block 13)Subroutine Searching for Non-Core Maximal Cubes (called by main program in Figure 1, Block 14) 114

Figure 4.4 Flow Chart of Subroutine Cube(called by Figure 2, Block 6 , or called by Figure 3, Block 12) 115

Table 1

LIST OF TABLES

Comparison of Different Approaches

1

CHAPTER IHISTORICAL REVIEW OF SWITCHING THEORY AND MINIMIZATION

1.1 Historical Review At the beginning, primitive mathematical ideas tend

to arise from the need of a better way to describe and analyze some special physical system; then through mathe­matician's hands it is further developed by deductive reasoning into a new branch of mathematics. Engineering tools for analysis and design of apparatus, starting from craftsman's rudimentary cut-and-try method to the borrow­ing of rigorous mathematics for refined and systematic synthesis, coincides with the progress of technological demands. In the 1680's, G.W. Leibniz' attempts to create a systematic mathematical logic led to the discovery of some of the basic Boolean algebra. 1 His contemplation of a calculating machine, although not rendered into hardware, was the starting point of modern digital sys­tems and computers. During the following 200 years, little progress was made in mathematical logic. In 1854,British mathematician George Boole published his work,

2"An Investigation of the Laws of Thoughts" , which mark­ed the beginning of modern symbolic logic and Boolean Algebra. Modern telegraphy and control systems have been using relay switching circuits since the late 19th

2

century and engineers and designers depended mostly upon experience and the art of trial and error to the design of switching circuits. In 1938, C. E. Shannon's introduction of the idea of using Boolean algebra in the analysis and design of relay contact circuits started, the rapid development of switching theory and digital systems.^

Thus engineers in digital or switching circuits acquired the proper tools to handle the analysis and synthesis of hardware: however, as the circuits and systems became more and more complex, the basic Boolean algebra manipulation became awkward and inefficient in handling the Boolean functions encountered in real sys­tems. The first systematic attempt at more efficient synthesis of logic circuits was made by Harvard Univer­sity's Computation Laboratory.5 The next important advancement was made by W. V. Quine in the early 1950's.6 In 1955, E. J. McCluskey generalized Quine's idea and also systematically developed the engineering notation and procedures for the synthesis of switching functions.

Another technique, the graphical method,also evolved. Veitch and Karnaugh , in slightly different manners inde­pendently derived a map from the Venn diagram which enables one to do the synthesis by visual inspection of the geometric patterns. Since then, there have been

numerous improvements in these two basic approaches. However, the covering problem in Quine-McCluskey's method and the limitation in the number of variables in the map method were always problems. The most recent papers offer some improvements and refinements in these areas. More detailed comparisons of various methods will be made in the next section.

In the late 1950's, R. P. Roth8 introduced the topological and simplex concept of treating Boolean functions, furthering the understanding of the inter­relationship between the Boolean expressions and Boolean functions. Roth also introduced the extraction algo­rithms which were the basis of the early computer algo­rithms developed at the University of Wisconsin9 and the computer programs at IBM.

As the technology in hardware improved, new devices and new circuits altered the cost value functions in minimizing hardware. Also, multi-valued logics, thres­hold logic, and sequential logic demands more generaliz­ed mathematical tools. With the coming of integrated circuits, the search for a minimum library of basic gates for all logic functions began. Thus a different approach to the synthesis of switching circuits develop­ed. New technology like MOS also required a different class of algorithms. Different approaches to switching

4

theory also arose from the increase in size of the inte­grated circuits and systems. The decomposition of Boolean functions, the symmetry of Boolean functions, and the multiple-output problem also became of more practical than theoretical interest. 10

With the increasingly realizable hardware devices for

threshold logic and multi-valued logic, newer and more generalized mathematical tools are in order for the practical engineer and designer.

By the late 1960's, Tison's Generalization of Consensus Theory and the computer algorithms derived from it dominated the industry; however, in the 1970's the need to refine the computer algorithms directed the synthesis of switching functions into a different horizon, that of faster and more efficient computation. The algorithms developed in Auburn University by C.C. Carrol's group started to look for relationships between minterms of a Boolean function, so that the covering problem could be avoided in the algorithmized methods. This approach conceptually coincides with the notion of realizing that synthesis can be done with a Karnaugh Map by recognizing geometric patterns. McKinney's directed search algo­rithm and Arevalo's Four-tuples Methods are recent contributions in this direction. More detailed compari­sons will be presented in the next section. A more

5

rational approach to the synthesis of switching functions is discussed in Section 3.

Footnotes: 1.1 Historical Review

1Bell, E. T., The Development of Mathematics (New York: McGraw Hill Company, 1962), pp. 553-554

2Boole, George, An Investigation of the Law of Thought (London: 1854).

3Bell, pp. 553-577, has a very good account of the historical development of mathematical logics.

^Shannon, C. E., A Symbolic Analysis of Relay and Switching Circuits, Transcript AIEE, Vol. 57 (1938) pp. 713-723. Other people in Russia and Japan had similar concepts, noteably, P. S. Ehrenfist of Russia in 1910, and Nakashima and Hanzawa of Japan in 1938. For more historical events, refer to G. C. Moisil, The Algebraic Theory of Switching Circuits (New York:Pergamon Press, 1969), and International Series of Monograps in Pure and Applied Mathematics, Vol. 41, pp. 11- 20.

5See Book ref. (12) which stemmed from simplifying vacuum tube flip-flop circuits, which procedure can beapplied to all switching functions

6See ref. Ql, Q 2 , Q37See ref. Kl, B78 See ref. B 6 , S4, S5, S16, S179 See ref. CIO, Sll, Si5

10 See ref. A 6 , Bl, CIO11 See ref. Tl

6

1.2 Comparison of Various Approaches

The basic concern of engineering design is to save cost and to improve performance. When we say cost, we have to consider both the software cost, hardware cost, as well as the difference in labor cost and the yield of constructing or manufacturing an item. When we say per­formance, we mean not only the performance of the spec­ified parameters but also the reliability.

To increase reliability, we have to increase redun­dancy which is the opposite of minimizing the circuits; to improve the tolerance of faulty hardware we also have to sacrifice minimization of circuits. Therefore, in search of an optimal mathematical tool for the analysis and synthesis of switching circuits, mathematical com­pleteness may not yield the most economical hardware, minimized hardware may not be worth the software cost of generating the design of hardware, and the most economical circuits may not have the desirable reliability.

(1) The Harvard Computation Laboratory Method 1

Listing all the minterms and all the possibilities of the basic functions which contain them is thorough and systematic; however, it is too bulky and cumber-

7

some to handle in any large size practical problem.

(2) The Standard Quine-McCluskey Method2

Using every minterm to build up a prime implicant table is also systematic and thorough, but suffers from the long process of enumeration and the very difficult problem of finding a minimum covering for the non-irredundant prime implicants. 3 Many other tabular methods 4 including R. P. Roth's cubical meth­od and Tison's generalized consensus methods are also influenced by this method. In fact, this meth­od presently represents the main stream of tabular or algorithmized methods.

(3) The Karnaugh and Veitch Map Method.Using grid squares to represent minterms and the relationship between minterms has both a graphical and a algebraic advantage; however, the graphical aspect of the method so dominates the significancy of the method, that its deeper algebraic implications are not fully developed. In this paper, some of the algebraic aspects of the method will be presented in a different manner. 5

(4) Boolean Algebraic Methods.R. T. Nelson's method6 of alternatively changing back and forth from products-of-sums to a-sum-of-

products during the algebraic cancellation or sim­plification of the Boolean expression has the ad­vantage of not needing to convert Boolean expression into minterms. But this method has little usage because of the number of variables it can handle. Nevertheless, the characteristic of not requiring expansion of the Boolean expression into minterms is certainly a desirable feature. There are other algorithms which have this property?

Topological and Simplex Cubical Method.Urbano, Mueller, and R. P. Roth, in a series of papers, applied the topological simularity of a n- cube and the Boolean minterms, and advocated the cubical complex methods. Subsequently, Tison ex­tended it to a generalized consensus method . 9 At the University, of Wisconsin a series of computer pro­grams were generated based on this method. This method contributes a better representation and notation for Boolean functions and a better visual­ization of the interelationship between minterms.The n-cube notation and concept is adopted by most authors in the field10 and we owe our notation and terminology to this approach.

9

(6 ) Graphical MethodsThere have been several instructive graphical meth­ods proposed in the last twenty years, noteably T. M. Booth's Vertex-Frame Method, M.E. Arthur's Geometric Mapping Method, and G.E. Marihugh and R.E. Anderson's H - Diagram Methods. 11 The H-Diagram, especially, has also proved to be fruitful in appli­cations to other areas . 1 2 In this dissertation we offer a new graphical representation. This approach illustrates the concept of graphical depiction of Boolean functions.

(7) Mechanical Devices Method(S22)A. Swoboda's contact grids ,F. B. Hall's Binary

(HI)Sieve , and A. K. Choudbury and M. S. Basu's Mechanical Charts represent attempts to use mechan­ical devices to achieve the synthesis of switching functioniC^] This approach demonstrates the feasi­bility of constructing a slide rule like device to synthesize switching functions. Combined with the characteristic of algorithm orientation, we believe a small hand-held portable calculator is also highly feasible.

(8 ) Computer AlgorithmsOriginally, the computer programs all stemmed from computerizing the Quine-McCluskey Method. There was

10

limited success in programming Karnaugh's Map Method. The following group of computer programs have been reported in the literature:

(a) IBM's R. P. Roth Group computer program based on R.P. Roth's Algebraic Topological approach!1 RH )

(b) C. C. Carrol, S. E. Jorden, W. A. Hornfeck,S. G. Shiva, and H. T. Nagle's Fast Algorithms at Auburn University.

(c) R. Slage, C. Chang, and R. C. Lee's sprouting algorithm from NIH.

(d) D. L. Dietmeyer and Y-H Su's Consensus Methods from Wisconsin University.

(e) M. H. McKinney's direct-search method algorithms at Texas A&M University.

(f) Z. Arevalo, J. G. Bredeson's Four-Tuble Algorithm.

14(g) Harmonic Analysis to generate prime implicant.

(h) Bantee, and Mott have some older computer pro­gramming which is of historical interest.

(i) Other Notable MethodsH. R. Hwa®*^, A. H. SheinmarfS^] S. K. Das and N.S. Khabra0 2 j . J. Mott^15) A. K. Cloudbury

11

Cl?and S. K. Das , D. M. Y. Chang and T. H.C2 Mott ,

Angelo R. MeoM ^ , J. F. GimpelG^ , N.N. Biswas®^D7S. C. DeSarkar, A. K. Basu, A. K. Choudbury ,

A. NatapoffN 1 , 0. L. Ostapko and S. J. Hong0^ .

In this dissertation, computer programs are given to illustrate the new general algorithm.One general characteristic is to economize com­puter memory and computing time.

(1) For detail procedure refer to ref. Book 12(2) Refer to reference Mcl, Mc2, M c3, Me4, or any

standard logic design and switching theory text book.(3) Many good methods of finding the minimum set are

theoretically interesting but not practical. References C 6 , C12, C15, C 2 , C3, P 7 , P8 , G 3 .

(4) See reference H12, S9, D2.(5) For original paper, refer to reference Kl, V2, For

examples and method, refer to a textbook, e.g. reference 4, 15, 36, 55, 39.

(6 ) Refer to reference M 4 .(7) Refer to reference S13.(8 ) Refer to reference VI, R 6 , R7, R 8 , R9, RIO.(9) Refer to reference T1(10) A good exposition of this notation and terminology

can be found in reference Ref. Book 19.(11) Refer to reference B7, A 5 , M3(1 2 ) lb Walch functions

12

(13) Refer to reference C 5 , S22, HI(14) Refer to reference Book 41

13

1.3 Desireable Characteristics and Discussion

A comparison of the approaches in synthesisizing Boolean functions and switching circuits indicates the following characteristics to be desirable:

1. Algorithm Oriented Characteristic

Because of the increasing use of computing machines and size, any approach to the synthesis of switchingcircuits has to be computerizable. F. Mileto and G.

_ (M8 ) (M9) . . . . . . .Putzolu' , did some initial work in estimatingthe merit and efficiency of computer algorithms for the minimization of Boolean functions based on the computing time and storage. However, because of the different criteria and concepts used in the different approaches, it was difficult to make an accurate and fair estimation. The size of the memory and the speed of calculation varies with different problems and settings. The skill of implementation into actu­al programs is also important. Although it is a very tedious job to evaluate the merits of the algorithms, it is mandatory to measure their effectiveness and efficiency. In this aspect, Roth's cube-array method is an improvement over Quine-McCluskey1s , and the consensus algorithm is another step forward. Each of the newer algorithms claim some order of improvement

on time and storage saving.

Limitation of the Size Characteristic

Most of the approaches do not differ much in efficien when the size of the problem is both small in number of variables and small in the number of elements of the on-set and don't cares. Therefore, the real measure of the effectiveness is on the size of problem that a specific approach can handle. The comparative merits have to be measured against each other with respect to a reference frame for size.The basic limitation of the map method is the diffi­culty of programming on the computer. Therefore it basically can be worked efficiently only with graphi­cal methods, and the difficulty of depicting 8 vari­ables limits its usefulness.

Visual Depictable Characteristic

The synthesis of switching circuits is not only to find a minimum circuit for a Boolean function, but also to observe the influence of changes of one part of the circuit on the remainder. It is not desirable to have to minimize the whole circuit every time we perturb the system. In this aspect, the map method offers the best opportunity. In fact, most of the

graphical methods try to gain this freedom and at the same time increase the number of usable variables.The H-Diagram is the most useful improvement since the original Karnaugh Map and can handle more vari­ables. The fact that H-Diagrams have been applied to the Welch function indicates that this is important, and the wide use of the Karnaugh Map in industry for smaller problems shows the desireability of this fea­ture .

Algebraical Structure Characteristic

Two-valued Boolean logic being the base for other logics, and logic design being the core of digital system and switching circuits, the algebraic struc­ture of an approach would not only enrich the switch­ing theory of Boolean functions, but could also be useful in other areas of switching theory.Table 1.1 shows a comparison of the major approaches mentioned in Section 1.2 with these four character­istics. In Chapter 2, we shall develop a structure so that we can maximize all the advantages of different approaches.

16

CHARACTERISTICS CHAR. 1 CHAR. 2 CHAR. 3 CHAR. 4(ALGO- (SIZE) (VISUAL) (STRUC-

APPROACHES________RITHM)__________________________ TURE)AKERS (Al) X XAREVALO (A3) X X XARTHUR (A5) XBATNI (B4) XBEATSON (B5) XBOOTH (B7) X XBOWMAN (B8 ) X

BREUER (B9,10) XBUBENIK (Bll) XCHANG (C2 & 3) X XCHOUDHURY (C4) XCHOUDHURY (C5) XCHU (C6 ) X

COBHAM (C7) X

CURTIS (Cll) X X

DAS (Dl) X X

DAS (D2) X XDUNHAM (D6 )GILL (Gl) X

GIMPEL (G2,G3,G4) X X X

GNAZALA (G5) X X

GOODRICH (G6 ) X

GORDON (G7) X

HALL (HI) X

HARRIS (HI) X

Table 1 Comparision of Different Approaches

CHARACTERISTICS CHAR.l CHAR. 2 CHAR. 3 CHAR.4APPROACHES (r Jt HM) (SIZE) (VISUAL) ^ u S

POLANSKY (P4) XPYNE (P7,P8) XQUINE (Q1/Q2,Q3) XREUSH (Rl) XRHYNE (R2) X XROBINSON (R3) XRODIN (R4) XROTH (R5,R11) X X XROY (R12) XSCHKOLNIK (S2) XSCHNEIDER (S4,S5) XSHANNON (S6 ,S7 ,S8 ) X XSHEINMAN (S9) XSHEN (S10) XSHIRA (S12) XSLAGLE (S13) X XSTOFFERS (SI5) XSURESCHANDER (Si8 ) XSVOBODA (S20,21,22) X XTISON (Tl) X XURBANO (Ul) X XVANDLING (VI) X XVEITCH (V2) X XVINK (V3) XZISSOS (Zl) X

T a b l e 1 I n n n ^ i n n o f l l

CHARACTERISTICS CHAR. 1 CHAR. 2 CHAR. 3 CHAR. 4(ALGO- (SIZE) (VISUAL) (STRUC-APPROACHES RITHM) TURE)

HARVARD (HI3) X X XHIRSCHORN (H6) XHONG (H7) X XHOUSE (H8) X X

HULME (H10) X XHWA (H12) XIKRAM (II) XKARNAUGH (kl) x XKARP (K3) X XLAWLER (LI,L2) X XLIN (L4) X XLUCCIO (L6) X XMC CLUSKEY (M3) X XMARCOVITS (M3) XMARIHUGH (M3) XMEO (M6) XMORREALE (Ml0,Mil) XMOTT (M14,M15) X XNATAPOFF (Nl) X XNECULA (N2) XNELSON (N4,N5,N6) X XOKADA (01) XOSTAPKO (02) XPARKHOMENKO (PI) XPETRICK (P2) X

Table 1 (continued)

19

CHAPTER II ALGEBRAIC STRUCTURE AND THEORY

2.1 THE MINIMUM MAXIMAL-CUBE COVERING

The basic notation and terminology of the switching theory in this dissertation can be found in the textbook "Mathematical Theory of Switching and Automata" by S. T. Hu, a well executed attempt to organize all the known re­sults in switching theory into a simple branch of pure mathematics with unified notation and terminology.^ To develop our concepts properly, we found further structure had to be constructed to facilitate the progress in some areas. However, our main interest is in the hardware used in computer and digital systems rather than in the pure mathematics. The format chosen is not of a pure mathematical symbolic-logic appraoch, nor of the engi­neering journal's approach, which goes directly from re­sults to examples placing definitions and proofs in the appendix. We chose to integrate the two. In this sec­tion, we will first discuss the properties of Karnaughmaps. Some of them have been utilized by other authors

(2 )m recent algorithms independent of our development.In following sections, we shall construct an algebraic structure to facilitate the use of those properties.

20

1. Adjacency:Quine-McCloskey's method demonstrates the importance ofthis property. Roth's and Tison's studies of the generalconsensus appraoch, expanded the use of this adjacency

(3)property, and most of the later authors also recog-(4)nized the importance of this characteristic. Recently,

f 1 1H. C. Curtis's adjacency table method , M. H.. M c 8 M c 9McKinney's directed search method, , and Z.

A3Arevalo's 4-tuple- Method ' all made much use of theproperty. Our appraoch is to use the number of adjacentminterms to any particular minterm as one of its charac-

(5)tenstic numbers. ' ' in the graphical methods, one of the advantages is that the adjacency can be determined by visual observation. In the analytical method, however, another kind of measure must be used to determine the adjacencies. The Hamming distance is the most logical metric and thus we shall adopt it. There are two partsof information pertaining to the adjacency. One is the number of cubes adjacent to a given cube; the second is the coordinates or directions of the adjacencies. In graphical methods, both are easily preserved except where there is a miscount due to human error. In our approach, we choose only to list the first part, the number of adjacent cubes, rather than identify each one of them individually, because we claim that the second part of

the information need only to be generated as needed.This saves labor of enumeration considerably.

2. Tops and Bottoms

Another important characteristic is the tops and bottoms of a cube. All the conventional approaches recognize this characteristic for any k-cube that may be chosen. However, their identification is still more or less a trial- and-error method as in the graphical approaches. We recognize the importance of this characteristic and will construct our algebraic structure in such a way that we can deduce the tops and bottoms analytically. Our in­vestigation shows that the tops and bottoms not only need the information of adjacency, but also the two new con­cepts that we introduced in the investigation, the degree of coverablity, and the order of coverablity for a specific minterm point in the on-set. These will be discussed indetail in the section. These concepts have been implied

S12in some of the recent results in a different form, that is, to identify the largest cube covering for a group of on-set elements. According to Hu's terminology, this is the maximal cube covering for a specific vertex in the on-set. Consequently, the minimum number of these maximal cube-coverings consists of the minimum set of prime implicants for any Boolean function. Our approach

especially emphasizes this point of finding the minimum maximal cube covering directly. At the start, it seems that we must identify the maximal cube expended from any two vertices. The difficulty arises from the lack of analytic ways to prove that it is indeed the maximal cube covering. Our effort of constructing an algebraic structure is due to this demand. After we achieved this goal, we found the algebraic structure we built has more implication to switching theory than was thought. This shall be treated in later chapters.

3. Interrelationship Between VerticesOne of the most important advantages of a graphical approach is that the detailed relationship of one par­ticular vertex with respect to all the rest of vertices of any on-set can be seen. In our investigation, it was found that from the concept of distance matrix and distance frequency matrix for the on-set, we were able to deduce most of these relationships easily. This new concept is a process similar to working with the map without drawing it. Additionally, we can employ most of the operation of the map method analytically.

4. Relationship Between CubesThe prime implicant table covering problem stemmed from the intrisic properties of the relationship between cubes.

23

The analytical approaches center their efforts in finding(7)efficient transformation of Boolean expressions' while

graphical approaches focus their attention on a closer(8)inspection of the graphical representation of the cubes.

There seems lacking a fundamental treatment of this aspect of the cubes. The cyclic cubes, the less-than cubes has haunted most of the researchers in switching theory since the day it was born. Our first approach to it is the theory of cube-graphs and branching graphs which will be fully developed in Section 2.3. Our second attack on it is in Section 2.4, where the geometric meanings of the cube-covering in the metric space (Qn , P) are introduced.In graphical approaches, the covering is drawn so that a minimum number of covering cubes can be found by trial- and-error, but no known theory has been developed to prove the validity of the existance theorem of a minimum set.We offer an attempt to prove the existance of such a minimum set with respect to a specific value function.

5. Beyond the Cube-CoveringWe constructed the metric space (Qn , d) out of the con- veniency for synthesising Boolean switching functions. It may well be worthwhile to look for more meaningful prop­erties of Boolean switching functions direct from this discrete space. Further work in generalizing it into

24

other kinds of switching functions is feasible and will be discussed in Chapter V.

Applications with illustrative examples for the Branching graphs and the geometric approach will be given in Chapters III

and IV.

25

Footnotes for 2.1

(1) Hu's book represented the kind of effort by an eminent pure mathematician we referenced in the opening statements of this dissertation (refer to Chapter I, paragraph 1.1, page 1 of this dissertation). Reference Book 19.

(2) Our initial results were submitted to IEEE Transaction on computer in December 1974. See repositore on Computer magazine. (Reference H13, H14)

(3) Adjacency is actually the basis for consensus.(4) Refer to Reference (H12)(5) We use column two of our characteristic matrix to

represent this characteristic.(6 ) We use (Qn , d) metric space. The definitions for

(Qn , d) and adjacency refer to Chapter II, para 2.2.(7) For example, the work of R. T. Nelson (Reference N4),

M. J. Gnzala (Reference G5) , F. Luccio (Reference L.6 ) , Petrick (Reference P2,P3), I . E . Byne and E. J. McCluskey (References P7, P8 ) and J. F. Gimpel (Reference G3).

(8 ) Our deep conviction that the process of finding an optimal covering by inspection on Karnaugh Map must be proveable led us to devise a way of gaging the distance between two cubes and then to determine how close they really are.

26

2.2. Algebric Structure of Minimum Maximal-Cube Covering Approach

Definition 1 : Qn is the cartesian product of Q setswhere Q = { 0.l). Qn is called a n-cube.

Definition 2 : (Qn ,d) is defined as a metric space withmetric function d being defined as follows:

If x = (xg' x2 ' xn ) / an^ y=(y1 »y2 »• •-yn )X &y E Qn , then: d(x,y) = ? = 1 |Xi“Yil is the distance between x and y. (The Hamming distance = the number of positions in which x and y differ)

Definition 2 and the definitions and theorems derived from it will suffice to render all the necessary information of a Karnaugh map. Although the main application of this approach is for the minimization of AND-OR (NAND) logic circuits. It can be shown that the theory developed in chapter 2 can be applied in many other areas.

Definition 3 : Let F= {a^,a2 , ...am } be the on-set ofa switching function F. The m x m matrix{A= [ a }] is defined as a^j=d(a^,aj). The i-th row of A will be denoted by d(a^,F). The m x m matrix B= [b^j]the distance frequency distribution matrix of F, that is, b^^ is equal to the number of points in F that have distance j to a.. We de­note by R(a^) the i-th row of B and by C(j) the j^column 0f

27

Definition 4 : Q = {1,0,*} , where * is the sign of don'tcare. If N= {l,2,...n} is the first N positive integer, then <j>:N ->Q*. The coordinate function defines all the coordinates of a k-cube in Qn . Furthermore d in (k)= the dimension of a k-cube = the cardinality of the subset (|> '*'(*) of N. and Cod(k)=n- dim(k) = the cardinality of the subset $ ^(Q) of N. 2

Definition 5 : A cube is covered by B cube if A £•. B.

Definition 6 : A maximal cube is defined as the largestcube covering a subset.When A cube in definition 5 reduces to a 0-cube, and B cube is taken from the union of on-set and don't care set, the definitions 5 and 6 are equivalent to the prime im- plicants in the Quine-McCluskey's minimization technique.

Example 2.2.1: f(x) = E (0,2,3,4,5,6.7,13) with on-set F = (0,2,3,4,5,6,7,13) then: d(0.2) = 1,

d(0,3)=2. etc.Thus Matrix A and Matrix B for F are obtained in

Figures 2.1 and 2.2 respectively. Figure 2.3 is the corresponding Karnaugh map.We notice from the Karnaugh Map in Figure 2.3 that Vertex 5 has 3 adjacent Vertices, 4, 7 and 13, and they are on the left hand, right hand and above Vertex 5 respectively. Similarly, from Row Five and Column 2 of Matrix B we know Vertex 5 has 3 adjacent vertices, and from Row 5 of

Figure 2.1 Matrix A Distance Matrix for Example

29

/ \

1 1 2 3 1\ /

13

Figure 2.2 Matrix BDistance Matrix for Example 2.2.1.

30

8 9 1 1 1 0

1 2

" I

1 5 1 4

4

1 5 7 I 6 1

0 1

3 1 2 1

F I G U R E 2 . 3 . K a r n a u g h M a p f o r E x a m p l e 2 . 2 . 1 .

Matrix A we know exactly they are 4, 7 and 13, because Column 4, 7 and 13 of Row 5 have values of 1. It is not known whether they are on its left hand side or right hand side. To find this, we have to compute them from 5 = (0101), 4 = (0100), 7 = (0111) and 13 = (1101), 4 differs at the fourth coordinate, 7 at the 3rd, and 13 at the 1st respectively. the relationship of 5 to any other vertices can also be derived similarly.

Definition 7 : <a,b> ^ is defined as the k-cube having aand b as its diametrical extreme points. It is also called a k-cube spanned by a and b.

Theorem 1: Let aeF and K F be a k-cube covering a. thenthere exists a unique point bek such that d(a,b)=k and K-<a, b>^= {xeF: d(a,s) + d(b, x)=k}

Proof: 1) Let b be the point in k obtained from a by com­plementing the varying coordinates of a. then d(a,b)=k and no other point in k can be at a distance k from a I

2) a: clearly <a,b = {xeF: d(a,x) + d(b,x)=k)b: In computing the integer d(a,x)+d(b,x) each

of the k positions where a and b differ makes a contribution of 1 to this number. Therefore if d(a,x)+d(b,x)=k, x must agree with a and b in the constant coordinate positions. Thus x e<a,b>^. -

32

Corollary 1 # {xeF: d(z,x)+d(b,x)} = 2 ^ (trivial)

Corollary 2: If a is a point in a k-cube in F, then thesum of the distance of x to any pair of diametrical extreme points is k.

Corollary 3: If a point a £ F is covered by a. k-cube inF, then there exists a k-cube in F with a as a diametrical extreme point.

Now we can examine example 2.2.1 again. Suppose the vertex 5 is chosen, d(5,F)= (2 , 3,2,1,0,2,1,1) . Since # (F)

3=8=2 and since at least one of the points (13), is not3-cube coverable, we consider a 2-cube covering. Let'slook for a 2-cube in F of the form <5 .b>2 that will cover5. By theorem 1, we consider vertices that are 2-distanceaway. In this case, vertices 0,3. and 6 are all candidates

2for b. By corollary 1 of theorem 1, we need 2 =4

where b=0,3,or 6 . We find d(5,F)+d(0.F) = (2,4,4,2,2,k 34,4,4). There are only three 2's, less than 2 =2 =4.

Therefore, we know < 5 .0 > 2 is not the 2-cube we are seeking nor is < 5 ,3 > 2 * However, < 5 ,6 > 2 does satisfy the require­ments. The only 2-cube existing is < 5 ,6 > 2 » Thus, the cubes can be found without the use of the Karnaugh maps.We trade off the immediate identifiable missing vertices of a cube to gain the advantage of algorithmizing the process.

33

Theorem 2 : (Qn ,d) and Distance Matrix A are equivalent tothe Karnaugh Map with on-set F.Proof: 1) By definition 2. matrix A gives us all the in­

formation about adjacency between elements of an on-set F.

2) By theorem 1, we are able to identify all the cubes on the map.

3) The physical relation of the map can be deduced from matrix A, and the n-cube coordinates re­presentation.

from 1) to 3), we conclude that we can generate the same kind of information from (Qn ,d) and matrix A as the K-map. Therefore, they are equvivalent. Q.E.D.

The minimization of Boolean functions will be demonstrated.Instead of using visual inspection, we can now use deduction.

2The disadvantage of this approach is that m units of storage is necessary to replace n squares of the map. Although

2matrix A is symmetrical, it still is necessary to use m /2 storage, where m is the number of minterms, and n is number of variables. However, in practice the memory is not the only factor, the calculation time is also important and matrix A does not have to be random accessable, it can be disc file .

In the next section, the theory for the synthesis of

34

switching functions will be developed. In Section 2.4, the geometric interpretation and additional theory for more efficient mechanization of the algorithms will be developed. Chapters III and IV will implement the algorithms for the results developed in Sections 2.3 and 2.4 respectively. Chapter V will extend theory into other areas.

Definition 8 : If there exists a cube that covers a pointin the on-set F of dimension j , this point is said to be j-coverable by a j-cube. If the point can be covered by

an i number of j-cubes, then it is j-coverable of i-th order.

Definition 9 : The subset F^ of F is the set of all thepoints in F that are j-coverable of order i.

Lemma 1 ; If a point is j-coverable, it is automatically (j-1 ) - coverable. (j>_1 )

Proof: If a is j-coverable, there exists a j-cube <a,b>^which covers a. Let c be a point in <a,b>j such that d(a,c) = j-1. Since (c<a,b>j) C's coordinates differ from a at j- 1 places while the remaining place it differs from b, there <a,c>^_^ C<a,b>j: <a,c>^_^ is a cube coveringfor a. Therefore, a is j-1 coverable.

35

k=j /^ik\Theorem 3 : Let = Min ( j ) - Then JL j is the maximal

number of j-cubes that may exist to cover the point a^ of the on-set F which is represented by the i-th row in matrix A and matrix B. The matrix L = [ Jt j ] represents the upper limit of the number of all maximal cubes that cover the points in the on-set F.

Proof: Since a cube is always symmetrical, the distancefunction is preserved under any rotation. Without lossof generality, the point I can be considered as the originof the cube. If the cube is of dimension j, then bycalculating the distance from every point in the cube tothe origin and determining the number of points with equaldistance, a distance frequency distribution pattern similarto the coefficient of binomial expansion (or Pascal triangle)is found. Thus, if there is a j-cube that covers the pointI, then the number b ^ must be greater than or equal to thecorresponding binomial coefficient (P) L 3 , and it is obvious

/ bik\that the most it can have is / \ of them.w

Therefore, for the range, k = o to k = j, the smallestnumber represents the maximum number of j-cubes that mayexist to cover a •. Q.E.D.1

Lemma 2: If a j-cube covering exists for point a^ thenthe number of points in F with 1 is at least 2^

36

Proof: Since it taken 2-1 points to form a cube, theremust be 2? points in the on-set F satisfying this condition.

Q . E . D .Theorem 4 : <a-a.>. is a cube of dimension j that cover

X J X

a^ and a^ if and only if, the i-th row added to the j-th row in the distance matrix A produces exactly 2 -1 entries of j's in the resultant row. Furthermore, these 23 entries correspond to the points contained in <a^,a^ >j and are the diametrical extremes.

Proof: 1. "If part":Since there are exactly 2^ points which have the

property of d (a± , a]c) + d(ak ,a. j) = j,it suffices to provethat any point k having this property is contained in thej-cube, <a-,a.>.. For any point ak having this property,

J Jg g gk differs from i or j by d(a-,a ) or d(a.,a.) in the

1 k K 3corresponding coordinates respectively. These places are all disjoint since any overlapping would imply that the distance between ai and a j is less than j , contradicting

g gthe fact that i and j are exactly the j-places in which ai and aj differ. Therefore, ak differs from aj at exactly those d(a^,ak) places where a^ and aj differ, and ak differs from aj at exactly those d(a^,aj) places where the re­maining ai and aj differ. Thus, ak is one of the points belonging to the j-cube with ai and aj as the diametrical extremes and d(ai ,ak ) + d(aj,ak ) = j. Since there are 2J

of them, it must contain every point of the j-cube,<a^,a^>

2 . "only if part":If <a^,a^>j is a j-cube covering ai,aj, then (a) d(ai,aj)= j, and (b) there are 2 points in <ai,aj >. includingai,aj. Now, if any point ak e <ai,aj>j, differs from aiin x places, then d(a^,aj<;) = x; if ak differs from aj iny places, then d(a, ,a.) = y. ak £ <ai,aj>, ak can notK 3differ from ai or aj other than where ai and aj differ.

• EL *Otherwxse k is not m <a^,aj>j. ..x+y = j = d(a^,aj),. .d^^a^.) = j and there are 2 -1 points in <I,J>. Thus,there must be exactly 2 -lentries of j's in the resultantrow formed by adding the rows a^ and a^ .

cl t** 2Corollary: Given i, j and k of the on-set F: If d(a^,a

+ dCa^aj) = d(a^,aj) = j, and if there exists any cube-£ a clcovering that includes i, j and k, then the smallest

cube is of dimension j .

Proof: If there is a covering, then d(a^,a^) + d(a^,aj)=d(a^,a^) = j implies that the cube I,J covering K is of dimension j . Any cube smaller than dimension j cannot cover I and J. Therefore <a^,aj> of j dimension is the

a cl clsmallest cube that covers i, j and k.

Definition 10: A maximal cube which has to be includedin any covering is essential and is called a core cube.

38

Lemma 3 : Any maximal cube that can be included in theunion of essential maximal cubes is redundant (trivial).

Definition 11; A maximal cube of a cover for a cube that can be included in the union of lower dimensioned cubes is

a less than cube, thus redundant.

Definition 12; A point is said to be a j-coverable core point, if, and only if, it has one and only one maximal cube covering of dimension j , with all smaller cube covering included.

Theorem 5 : If x is a j-coverable core point, then thereare exactly j points in the on-set having distance of 1

from x.

Proof: Trivial (By definition 12)The inverse of this theorem is not necessarily true.

Footnotes for 2.21. Reference book 18, 19.2. Reference book 19, page 63-65

39

2.3 CUBE GRAPH AND BRANCHING GRAPH

To facilitate the analysis and synthesis, the con­cept of j-coverable is introduced so that the on-set F of a switching function can be divided into disjoint subsets of j-coverables. To further improve the ability to identi­fy and select the cyclic cubes and the less-than cubes, the concept of a candidate for a core point, and the core point as well as the cube graph and a branching graph are used.

Definition 13: Two points are connected, if and only ifthey are covered by a common point.

Definition 14: Two cubes are connected, if and only ifthey both cover at least one common point.

Definition 15 ; A j-coverable branching graph is a linear graph representing all the connected j-coverable points.

The nodes in a j-coverable branching graph represent the diametrical extremes of the connecting cubes and the branching representing the connecting cubes.

The initial node can be any point in any set of connected j-coverable points. The next nodes are those points forming a connecting j-cube with the present node such that the diametrical extremes are the present node and the next node.

The terminal nodes are: (1) the nodes that have onlyone branch connected to them; (2 ) the nodes whose remaining

connecting branches going to the new nodes consist of cubes already represented by existing branches; (3) the nodes belonging to don't care or higher coverable sets.

Definition 16: A j-coverable cube-graph is a linear graphrepresenting the j-cubes covering the connected j-coverable points.

Definition 17: Two j-coverable branching graphs are saidto be equivalent if and only if they have the same j- coverable cube-graph.

Lemma 4 : For every j-coverable branching-graph, there isone unique j-coverable cube-graph.

Proof: Let G be a j-coverable branching graph with C andC 1 being cube-graphs of G. If C is not coincident with C', then there is at least one of the cubes which is different. Since every branch in a branching graph re­presents a distinct cube, the C must coincide with C'.

Lemma 5 : A set of connected j-coverable points has a unique cube-graph.

Proof: Let Fj be a set of connected j-coverable pointswith C and C' being the cube graphs of F^. Let cube c^Cand c^C1. There is then at least one point x in c thatis not in any cube of C'. Since x is not in any of thecubes in C 1, x is not connected to F ^ . This contradictsthe fact that x is a member of F ..3

41

Theorem 6 : A set of connected j-coverable points must have j- 1

2 different but equivalent branching graphs.

Proof : Since there are 2^ ^ different pairs of diametricalextremes to a j-cube, there are 2 ^ different pairs of end nodes for each branch of a branching graph. When joined, there are 2^ ^ distinct branching graphs. By Lemma 4 and Lemma 5, all are equivalent.

Theorem 7 : The minimum set of non-redundant maximal cubesis the union of minimum covering for the branching graph.

Proof: Since each branch of a branching graph representsa distinct maximal cube, the union of the minimum covering for each branching graph of a j-coverable subset (j = 0 to k) is the minimum set of non-redundant maximal cubes.

This theorem transforms the process of finding a minimum covering for a table of prime implicants into the finding of a minimum covering for each branching graph of a j-coverable subset of the on-set, taking the union over j = 0 to k, the highest dimension of the maximal cubes.

The advantages are: 1) No redundant maximal cubes are generated; 2) The on-set is paritioned into subsets of j-coverable connected points so that the covering problem can be treated separately for different subsets; and 3) The branching graphs are easier to work with.

The following theorems relate to the covering of branching graphs.

42

Theorem 8 : A minimum covering of a branching graph isthe sum of the least number of branches needed to cover every circuit in the branching graph.

Proof: Since a branching graph represents the maximal cubes covering connected j-coverable points, and each loop in the branching graph represents a set of cyclic maximal cubes, therefore the sum of the least number of branches needed to cover each loop is the minimum covering of a branching graph. An open loop is a set of incomplete cyclic maximal cubes.

Theorem 9 : The least number of branches needed to covera loop of n branches is ^ if n is even or is ( j + 1 ) if n is odd.

Proof: Let the loop have n branches, then there are nnodes. Therefore, at least one half of these branches is needed to cover all those nodes because a loop is a closed plane curve and, within a loop, each node has exactly two branches connected to it. If n is odd, since a half­branch can not be used, an additional branch must be used.

Corollary 1; If the number of branches in a loop is even, the minimal covering of the loop is found by choosing every other branch.

Proof: Since in any loop, a cube represented by a branch can be divided into two parts: 1 ) the common part of

43

this cube and the cube represented by the preceding branch; 2) The other is the common part of this cube and the cube represented by the following branch. Therefore if the preceding and the following branches are chosen, then the cube represented by this branch is covered and thus is redundant. Q .E .D .

Corollary 2 : If the number of branches in a loop is odd, then except for the starting or finishing of the covering which has to be two consecutive branches, every other branch is the minimum covering.

Proof: Since the number of branches is odd, every other branch can not be chosen to cover a loop. There must be one part of a cube covered by two consecutive branches. Thereafter, it is the same as Corollary 1.

Theorem 10: If there are terminal nodes in a branching graph of the second kind, then these nodes have to be treated first for the elimination of the higher-dimensioned less-than cube.

Proof; If a terminal node is of the third kind, then it must be a higher-dimensioned cube coverable point. Since it is a terminal node of a branching graph, it has only one branch connected to it and that branch is necessary unless it is a less-than cube of some lower dimensioned cubes. Therefore, the higher dimensioned cube which also

44

covers this node may not be necessary, if the rest of the cube is covered. Thus, the set of all terminal nodes of the second kind must be tested for redundancy of the higher dimensioned maximal cubes in a higher coverable branching graph.

Theorem 1 1 : If a terminal node of a branching graph is ofthe first kind, then it is a core point, and the only branch connected to it represents an essential maximal cube.

Proof: Since a terminal node of the first kind by definition has one and only one maximal cube covering it and since no higher cube covers it, therefore it meets the definition of a core point. The only branch connected to the node re­presents the maximal cube mentioned above. Therefore, it is essential.

Theoreml2: The more loops sharing a branch, the higher isthe priority for choosing this branch.

Proof: The more loops sharing a branch, the more thisbranch can be used to cover different loops. Since this branch represents a partial covering for each loop, a smaller number of branches is needed. Q.E.D.

If there is more than one branch connected to a node, and no branch is shared by two or more loops, then the priority for choosing a branch for this node is inversely

45

proportional to the number of branches connected to the node.

Definition 18: If two branching graphs of differentdimensions are linked such that the branches connected to the linking node in the higher dimensioned branching graph can be omitted in a minimum covering, the linkage is inseparable.

46

2.4 Geometric Structure of Maximal-Cube Covering

Since the use of Venn Diagrams, graphic depictingand geometric representation has been widely used inminimization of switching functions. In the author'searlier work, the geometric structure of the Karnaugh-Map and generalized finite discrete point representation

H14 H15of Boolean functions had been discussed. ’ Thebasic aim of using the geometric structure to represent functions is to seek visual recognition of the complex interrelationship. The analysis by the geometric struc­ture of Maximal-cube covering approach is different than the graphical means.

(a) Diametric ExtremesThe importance of using diametric extreme terminology

is that cubes can be treated as if they were circles or spheres. Thus the drawing of circles tangent to each other would be a way of finding the minimum covering. Another advantage of this terminology is that the sense of interior points, boundary points, and exterior points can be introduced. In the case of Maximal-cubes of Boolean function, there are boundary points and exterior points, but no interior points. If x, y are a pair of diametric extremes of cube <x, y >, then any point z is either on the boundary of the cube or outside the cube.

Another feature of the diametric extreme is that the

47

absolute origin of corrdinates is not needed, because any pair of diametric extremes can be used as the refer­ence .

(b) AwaynessThe major advantage of graphical methods is that the

distribution of cubes (a minterm point can be treated as 0-cube) can be visually observed. However, pattern re­cognition is always needed. The purpose of introducing the concept of awayness and aloofness in the maximal-cube covering approach is to simulate the kind of pattern re­cognition used in the map method.

To arrive at a numerical measure of the awayness or closeness of a cube to the existing cube covering, the concept of distance function is expanded.Definition 9 : A k-cube is re-defined as a pseudo-circlewith diameter of k distance units.Definition 20: The diameter of a pseudo-circle is thelongest span of a pseudo-circle[equal to the dimension of a cube].Definition 21: The two end-points of a diameter of apseudo-circle is called the pair of diametric extreme points, or the diametric opposite points.Definition 22: d(x, A) is defined as the distance fromvertex x to any pseudo-circle A (cube A ) . d(x, A) = Min ( d(x, a ) : aeA)

48

Definition 23: D(A) is defined as the distance units ofdiameters of pseudo-circle A (cube A) D(A) = Max {d(a^, a2): ai, a 2 e A}Definition 24: d(A, B) is defined as the distance betweentwo cubes A and B.d(A,B) = Min {d(a^, a2) : a. , eA,Theorem 13: The distance of a point x, the exteriorpoint of a pseudo-circle (cube) to the pseudo-circle is the number of changes of coordinates between x and any point inside the pseudo-circle excluding those places where the points of the pseudo-circle vary their coordi­nates among themselves.Proof: By Definition 18, d(x, A) = Min {d(s, a) : ae A}

Let L = m + n = d(s, a) for all aeA.m = the number of changes of coordinates between

x and a at those places where the pseudo-circle(or cube) A varying coordinates among its points.

n = the number of changes of coordinates between xand a, where the coordinates for all points inthe pseudo-circle (or cube) A are invariant.

Then n is constant for all points in A. Only m varies.Therefore: d(x,A) = Min {d(x, a) : aeA)

m = k = Min (m + n) = n.

m = 0Q.E.D.

49

Corollary 1 : The distance from interior points of pseudo­circle to the pseudo-circle is zero.

Proof: As in the proof of Theorem 13, when n = 0.

Definition 25: The smallest cube or pseudo-circle havinga point of a cube A (or pseudo-circle A) and any point x is defined as the connecting cube (or pseudo-circle) between A and x.Definition 26 : The smallest pseudo-circle (or cube) havinga pair of diametric extreme points where one is from pseudo­circle A (or cube A) and the other is from pseudo-circle B (or cube B) is called the connecting pseudo-circle (or cube) between pseudo-circle A (or cube A) and pseudo­circle B (or cube B) .Theorem 14: d(x, A) = the dimension of the connectingcube between x and A,d(A, B) = The dimension of the connecting cube between

cube A and cube B. (Since pseudo-circle = cube, we will use cube from now on.)

Definition 27: The foot of a connecting cube between a cube A and a point x is a point in A such that C = <x,a>

Theorem 15; The distance between a point to a cube is equal to the distance between the point to the foot of the connecting cube.Proof; Let a be the foot of the connecting cube for point

50

x to cube A. By Definition 27, d(a, x) = dimension of connecting cube. By Definition 25, d(a, x) = Min {d(x, y): yeA} = d (x,A) Q.E.D.

Theorem 16 : If a and b are a pair of diametric extremes,

then all the places of the coordinates where the points of the cube vary their coordinates among themselves are complementary.Proof; Let <a, b> be a cube of k dimension, then there are k places among the coordinates where the points of the cube vary among themselves. Since d(a, b) = k and only k place can vary, therefore a and b have to be complementary at these places. Q.E.D.

Lemma 6 : If a^, a 2 are diametric extremes of a k-cube,then d(ai, a.2 ) = D(A) = k = dimension of the cube = the diameter of the sphere or circle. (Trivial)

Theorem 17; If a^, a 2 are diametric extremes of a k-cube and xeOn , then

(1 ) d(alf x) + d(a2 , x) = d(a 1 7 a 2 )=k xe <a^, &2>k*

(2 ) d(a^ x) + d(a2 , x) < d(a^, a2)~* x is outside of

<al a 2>k

(3) d(a^, x) + d(a-2 , x) < d(a^f a?) does not exist.

Proof: . (d, On ) is a metric space, impliesd(alf x) + d(a2 , x) >_ d(alf a 2 ) •

by Theorem 1, Corollary 1 (1) holds.(2 ) implies that point x has some coordinates differing from a- , or a2 which are not among the places ai and a2 are different. Therefore x is not included in <a^, a 2 >. Thus x is outside of <a]_, a 2 >. Q.E.D.

Theorem 18: If A = <a^, a 2 >k, then for any x£Qnd(ai, x) + d(a2 , x) - k = 2 dwhere d = d(x, A) = the distance from x to A.

Proof: If x=4> (Trivial)

If x?^ / d <ai' x) + d (a2 ' X) > d(alr a2) = k (by definition of metric space)

1) If x a<a-^, a2>k then d(x , A) = 0 By Theorem 1, Corollary 1;d(a^, x) + d(a2 , x) = kd(ai, x) + d(a2 , x) - k = 0 = d(X,A)

2) If xe< al» a 2 > k/ then By Theorem 13;d(a^,x) = Mi + d(X,A) ....(1) d(a2 ,x) = M 2 + d(X,A) ___ (2)Where d(x,A) = The number of coordinate positions

where x differs from A, among the sub­set of coordinate positions where A does not vary its coordinates amongits elements.

52

= The number of coordinate position x differ from aj_, among the subset of coordinate positions where A varying its coordinates.

M 2 = The number of coordinate position x differ from a2 ,

among the subset of coordinate positions where A varying its coordinates.

Since a^ and a 2 are complementary at the varying coordi­nate positions, Mi and M 2 must also be complementary.Thus Mj + M 2 = k.

(1 )+ (2 )resulted ind(a^, x) + d(agf x) = Mi + M 2 + 2d(X,A) d(a-j_,x) + d(a 2 , x) - k = 2 d.

Q.E.D.Since <a^, & 2 > is an arbitary pair of diametrical extremes, and x is any point in Qn , therefore, this is true for any pair of diametrical extremes.Thus, d(a^, x) + dfa^/ x) - k is an invariant.

Corollary 1 : For all xe Qn a e A .d (xi a) = M + d(x, A)

Where M is the number of positions x differ from a, among those coordinate positions A is varying, and d(x,A) is the distance of x to A.

Proof: See main theorem.

Corollary 2: For all xe Qn , aieA, a 2 eA. if {d(a^,x) -d(x,A)> + (d(a2 ,x)- d(x?A)} = k. a^, form a pair

53

of diametrical extreme points of A.Proof; See theorem 1 8 (gives an invariant which is going

to be useful in mimization. It will be called away­ness .)

Definition 28; Awayness is defined as the distance of a point to cube, wak(s) = d(a,x) + d(b,x) - k = 2 d Definition 28 can also be extended to the distance between two cubes A, B.

Definition 29: Awayness of two cubes A, B is defined asWa (B) = d(a1# B) + d(a2 , B) - kA + # (R)where kA is the dimension of cube A, and R is the sub­set of coordinate positions when both A, and B are

varying its coordinates. We will define # (R) as the degree of skewness.

Definition 30: The skewness is defined as the number ofcoordinate positions both cube A and cube B are vary­ing. We will denote it by S (B ) = # (R) = SKg (A) =SK(AB)

Theorem 19: Awayness is independent of drametric extremes

Proof: (1) If one of the cube is a 0-cube then theorem 19reduces to theorem 18.

(2) If cube A and cube B are not 0-cubes, then: Wa (B) = d(alf B) + d(a2 ,B) - KA=2d-#(R)

= d(bx ,A) + d(b2 ,A) - Kb = Wb (A)

where <alfa2> RA=A, < b 1 ,b2 > KB=B •

d is the distance between A and B. # (R) is the number of common parts in the varying coordinates of cubes A, B.There are four different categories of coordinatepositions for cubes A,B denoted as follows.p = those positions only cube A varying its coordinatesQ = those positions only cube B varying its coordinatesR = both A and B varying its coordinates;S = neither A or B varying its coordinatesIt is obvious # (P+R) =

#(Q+R) = Kb# (P+Q+R+S) = n, the dimension of Qn .

Let d(x,y) = (xeA, yeB} .Then d(x,y) = M + M + m + M„'2 ' p q r s

where = The number of the positions X and y differ in values among P.

Mg = The number of the positions x and y differ in values among Q.

Mr = The number of the positions x and y differ in values among R.

M g = The number of the positions x and y differ in values among S.

We observe the following facts.

55

M = Constant, because both cubes A, B are not varying.Mp varies from 0 to I<A - # (R)Mg varies from 0 to Kg - # (R)Mr varies from 0 to # (R)Thus d (a,b) = d(A,B) = min d(x,y)

= MsThat is, the closest points a, b in A and B only differ in those places neither A and B are varying. If we comple­ment a, and b in A and B, we obtain c in A, and d in B.and by definition <;c,a> = A, <b,d> T.n = B.

J\A i\J3

Furthermore d(c,d) = K,+K -2A(R)+M , because in the pro-A o Scess of complementing a, and b,we change only those P and R values for a, and Q and R values for band none in S. Since we are simultaneously complementing R for a and b, there is no difference in the resultant c, and d. Therefore none con­tribute to distance number; while in S subset; c and ddiffer the same as a and b, thus contribute M . However insthe P and Q subsets, c and d differ #P and #Q as against no difference for a and b. Totalizing all the differences, d (c,d) = #(P) + #(Q)+Ms .. d(c,d) = KA~# (R)+Kg-# (R)+Mg

= Ka +Kb - 2 #(R)+MS

(These are the most distance points among A and B.) i.e. d(c.d)=Max (d(x,y) | )(eA' yeB}

d(a,b) = M s=Min d(x y ) ........... (1)d(c,d)=KA+KB - 2#(R) + M s . . . . (2)

56

for any pair of diametric extremes a^ , a 2 of A.Thus, d (a1 ,b)+d (a2 ,b) = KA+2d(b B ) ........... (3)d(a1 ,d) +d(a2 ,d)=KA +2d(d,B) ............. (4)

Since 2d(d-^B) = d(c^b) + d(a-^b) - KA ......... (5)2d(d1 B) = d(c1 d) + d(a1 d) - KA ........ (6 )

and Since c, and d were obtained from a, and b,d(c,b) = #(P) + # (R) + M g = Ka + M s .......... (7)d(d,a) = #(Q) + #(R) + M g = KB + M g .......... (8 )Substituting (1), (2), (7), (8 ) into (5), (6 )2 d (b,B) = KA + M s - KA = 2 M s ........ (9)2 d(d,B) = k a +k b - 2 #(R) + m s+k b + m s - k a

= 2 Ck „+M= - # (R) 1 (10)£5 SSubstituting (9) and (10) into (3), and (4).d(a1 ,b) + d(a2 ,b) = KA + 2 M g ... (11)d(ax ,d) + d(a2 ,d) = KA + 2 Ekb+Ms - #(R)] (12)

add (1 1 ) (1 2 ) and rearrange the left hand side to obtain.d(a1 ,b) + d) + d(a2 ,b) + d(a2 ,d)= Kb+ 2d(alfB) + Kb + 2d(a2 ,B)= 2 [Kb + dfa^B) + d(a2 ,B)]

While the right hand side is2 [Ka + Kb + 2 M s - #(R)]

KB + d (a1 ,B) + d(a2 , B)

= KA + kb + 2 M s - # (R).-.d(alfB) + d(a2 ,B) - KA

57

= 2Ms - #(R) = 2d(aib) - # (R)

= 2d - # (R)

Similarily WB (A) = 2d - # (R).Q.E.D.

Remark: If one of the cubes reduces to a point (i.e,0-cube), then # (R) = 0. Otherwise, the 0-cube becomes a higher dimension cubes. And from theorem 19, a new para­meter appears, that is the # (R) which gives the relative direction of two cubes, which can be observed on the map. Theorem 18, 19 are analogous to plane geometry theorems shown in Figure 2.4.1, and Figure 2.4.2. Other examples and demonstrative figures are given at the end of this section and in Chapters IV and V.

For Figure 2.4 We have:Bx.Ax = Dx.Cx

= (D+d) (d)2= Dd + d = invariant

for any DCx line through x which intersects the circle

58

Figure 2.4 The Analogy of the Awayness of a Point to a Cube in Plane Geometry

F

D

Figure 2.5 The Analogy of the Awayness of a Cube to a Cube in Plane Geometry

59

For Figure 2.5.If, Let AC = BC = 3s dThen GC.FC = +% d 2 ........... (1)

IC.HC = d +% d 2 ........... (2)d d 2Gc.Fc + Ic.Hc = j (D1 +D2) + j

GC.FC.IC.HC = (D1 + |) (D2+|) j

For minimizaing minterms of a switching function we have the following theorems.Theorem 20: If a point is j coverable and having awaynessof Wab to cube <a,b> of D dimension, then the number ofthe common points between the cube covering of the point

n j. j-W

and the original cube is zero if Wab is greater than j .If Wab is smaller than j, then there are at most 2 J afc> points in common between the old cube and the new one.Proof; If Wab is greater than k, then x is aloof from the old cube; thus no common points exist. If Wab is smaller than j, then there may be at most 2? Wab common points.

It is clear that Wab is a very important factor in determining the priority of the Multiple-coverable points. Theorem 21: j-wab alwasY smaller than D, the dimensionof the chosen cube.Proof; If j“Wab is larger than D, it is possible to find a cube that covers x and the whole cube <a,b> of dimension

60

j. But <a,b> is the largest cube to cover a and b. If the j cube does not intersect <a,b> then they are sepa­rated and there is no relation. This can also be extended to higher ordered cubes. Let us define the maximal distance of two cubes as the envelope of two cubes.

Definition 32: The envelope of two cubes A and B,[e(A,B)] is defined as e(A,B) = K^+Kg - # (R) + d(A,B)

Theorem 22 : The smallest cube covering both A and B hasthe dimension of e(A,B) + # (R).

Proof: From the proof of Theorem 19.e (A,B) + # (R)

= Ka +Kb - 2 # (R) + d(A,B) + # (R)= -((Ka - # (R)} + {Kb -#(R)}+ d (A,B) + # (R)

where KA - # (R) = the number of coordinate positionsonly cube A is varying.

Kg - # (R) = the number of coordinate positions only cube B is varying

# (R) = The number of coordinate positions both A.B are varying

d(A,B) = The number of coordinate positions neitherA nor B are varying but where A and B differ.

Q.E.D.Theorem 23: If cube A and cube B are of the same dimensionR, and d (A,B) = e(A,B) then all varying coordinates coin­cide.

Proof; e(A,B) = K+K-2#(R)+d(A,B) (By Theorem 22).'. 2K - 2# (R) = e (A.B) - d (A,B)

=0

61

K = # (R) Q.E.D.

Corollary: Of e(A,B) = d(A,B) = 0, A=B.

Theorem 24: If cube A and cube B have the same dimension, d(A,B) = 1, e(A,B) = K+l then A and B form a(k+l)-cube.

Proof: e(A,B) = K+R - 2 # (R) +d(A,B)= 2K-2#(R) + d(A,B)

By Theorem 22, the smallest cube covering A and B has a dimension of

e (A,B) + #(R) = 2K - 2# (R) + 1 + # (R) = 2k-#(R)+l#(R)= 2K+1 - e(A,B) = 2K+1 - K - 1

= K.e (A,B) + #(R) = 2K - # (R) +1 = 2K-K+1

=K+1. Q.E.D.

Theorem 25: Let f(A,B) ~ max (d(x,y):xeA,y^B}.

Then (A,B) = e(a,b) iff#(R) = 01 ) if part,

Proof : f(A,B) = Max (d(x,y) : X£A, y £B) .= Max {M + M + M + M .}p q r s

Maximum of m = K,- # (R) Maximum of M =K„ - # (R),P ** CJ .DMr= # (R), and Ms= d(A,B)

Thus f (A,B) = KA-#(R) + Kb -#(R) + #(R)+d(A,B)= KA + KB " 2 # (R) + d (A, B) + # (R)= e (A, B) + #(R)

/. # (R) = 0

62

2 ) only if part.# (R) = 0 . implies M =K,,M =K_ M = 0P CJ £> T

M s = d(A,B)4

. . f(a.b) = Ka + KB + d(A,B).e(A,B) = KA + Kg - 0 + d(A,B) = K^+Kg+d(A,B)

.. f(a,b) = e(A,B)

Corollary; f(a,b) = e(A,B) + SK(A,B)

Theorem 25 : Given cubes A, B, and d (A,B) = 0, thenthe number of common points = 2 = 2 SKM B )

Proof: (1) If d(A,B) =0, there are no coordinate positionsdifference outside of those position where either A, or B are varying. Thus the intersection of cube A and cube B has a dimension of #(R)=SK(AB). and their varying coordinate positions are exactly those # (R)

positions where both A and B are varying.

Theorem 26: Given cubes A and B. and d(A,B)=l.mhan cv f ktj uKA+KB k A+k B +2e(A,B)-l. Then SK(AB)=— -— , f(a.b) = — -----

Proof. e(A,B)=KA+KB - 2SK(AB)+d (A,B).'. Ka +Kb - 2SK (A,B) = 0.'. Ka +Kb = 2SKSK(AB) = KA + KB

2

f (a,b) = e (a,b) + SI<(A,B) = 1 + KA+KB = 1Sa +|b.+1Q.E.D.

63

(c) AloofnessAnother measure of the merit in finding the minimum

covering is how evenly can the largest cubes be arrang­ed. Ideally, we would like to find a maximal-cube

adjacent to the existing cube covering. For this, we shall introduce the concept of aloofness, which turnsout to be a way of gauging how evenly we can arrange themaximal cubes. In order to define aloofness, we shall generalize the definition of adjacency among 0 -cubes of any dimensions.

Definition 33: The "aloofness" is defined as the frequencythat a point's awayness exceeds the highest dimensions, that is, the number of times when W , (x) - d > 0 , dvcU3 X — Xbeing the highest dimension of cubes that x can have formaximal-cube covering.

Definition 34: W*(B) - d„ > 0 is the aloofness for cubeB with regard to cube A.It us evident that the selection of the next point should be based on smaller awayness and larger aloofness so that we can evenly spread out as will be aemonstrated in chapter IV.

(d) AdjacencyThis property is commonly used in literature, and

refers to finding as many adjacent sub-cubes to the exist­ing cube covering as possible.

64

Definition 35: Adjacency is defined between two cubes only if cubes A and B have a distance of 1, i.e. d(A,B) = 1.It is obvious that we want to choose the maximal cubes that are adjacent. The reason t h e ’map method is so power­ful is that one can observe this by visual investigation.In the method given here, it is analytical, not a trial and error method. To achieve the indicated results, a merit value is assigned to each uncovered point or cube.

Definition 36: The degree of adjacency is defined as thenumber of distinct sub-cubes that are adjacent to the existing maximal cube covering.

e) Probability Overlapping.In the process of choosing maximal cube covering,

the existing maximal cube covering should be overlapped as little as possible.

Definition 37: The probability of sharing common partsof a new cube with the existing cube covering is: (share) =D,if the highest coverable maximal cube has a dimension of J > D + Wab or = J-Wab,if J I D + wab.

If the awayness and the aloofness are the same, then the next criteria is the relative probability of sharing.The last check is to compare the real over-lap of common parts.

65

Definition 38:ifd(A,B) > 1 for all B C F, A is isolated.

Definition 39: Let * = 1/2 to be the don't care valueQ* = {0, 1, *} for the value of the coordinates of the center of pseudo­circles .

f) Other related structure.Lemma 7: if d(A,B) = 0, then A,B are coincident Lemma 8 : If d(A,B) > 1, then A,B are separated

66

CHAPTER III

BRANCHING GRAPH APPLICATION AND ILLUSTRATIVE EXAMPLES.

3.1 THE ALGORITHMThe minimization of two-level sum-of-products logic

can be summarized as follows:Step 1: The distance Matrix A is constructed;Step 2: From Matrix A, the distance-frequency

distribution Matrix B is obtained;Step 3: From Matrix A and Matrix B, the limit matrix

L is obtained;Step 4: From the theorems and matrices A, B, and L

and with possible core points as candidates,the essential maximal cubes (i.e, essential

? 1prime implicants) U M. are selected byj=o -1

starting with the highest dimensioned j-cubesP 1

down to the lowest, i.e., finding U M^(f) ZDj = 0

U F^ (All of the essential maximal cubes have 3been found) ;

Step 5: The core points are used as the initial nodesto find the branching graph for each set of connected j-coverable points, where j=k, k-1 ,k-2, ....... 3,2,1 and k is the highestdimension of the maximal cube covering;

Step 6 : If there are terminal points of higher cover-

67

able points or branches containing higher coverable points, then they are compared with the cube graph of higher dimension, to see if any less-than cubes exist;

Step 7: A minimum covering of the branching graph,which the minimum non-redundant set of max-

p^q ±imal cube covering, i.e., U M M F . is found.i=l 3 j= 0

3.2 Demonstrative Electives:Example 1: To illustrate that a non-essential prime im-

plicant will not be generated if it is already included in the set of essential cubes:

f (X) = Z (0, 2, 3, 4, 5, 6 , 7, 13) .Step 1: Construct Matrix A (Fig. 1). (Matrix A is

symmetrical).Step 2: Construct Matrix B (Fig. 2) from Matrix A.Step 3: Construct Matrix L (Fig. 3) from Matrix B

1) Since only four points (4, 5, 6 , and 7)satisfy the necessary conditions for a 3-cubecovering, no 3-cube covering exists. Multiple 2 -covering may exist for these points.2) Only points 0 and 3 are possible 2-coverable core points.3) Point 13 is a possible 1-coverable core

point.

68

Step 4

Step 5

Step 6

: Since 0 and 3 are of same order and degree,point 0 is arbitrarily chosen first. Only 2- cubes are being dealt with in this case, and therefore any distance over 3 is immaterial. All distances 3 or over are replaced by X.The table in Fig. 4 shows the distances for 0, 3, 5, and 6 taken from Matrix A. Since 3, 5, and 6 are points having a distance 2 to 0 , they are possible candidates for diametri­cal extremes. Theorem 2 shows that only the <0 ,6 > cube exists. (<0 ,6 > = 0 ,2 ,4, 6 ).Now L is updated to L- , by changing the row representing 0 in L to (-,-,e), indicating that point OeF^, and 2, 4, and 6 belong to the don't care conditions (Fig. 5)

: Continue to point 3, which is another candi­date for a 2 -coverable core point and find the <3,6> which is an essential 2-cube. Up­date to L 2 (Figs. 6 and 7).Having exhausted the 2-coverable core points, the possible 1 -coverable core points are considered. Thus, point 13 is to be covered by <13, 5> . Update L2 to L 3 (Figs. 8 and 9). Since all points are covered, the process is complete.

69

13'

/0 1

6 7 13\

3 4 3 2 1 3 2 0

\ /

Figure 3.1 Distance Matrix A of Example 1

70

P0 Pi P2 P3 P4\

2 0

1 0

1

1 1 2 3 1\ /

13

Figure 3.2 Distance Frequency Matrix B of Example 1

71

13

c0 C1 c2 c3 c4

/ \

\

0 0

0 0

0 0

/

Figure 3.3 Matrix L of Example 1

72

0 2 3 4 5 6 7 13

0 0 1 2 1 2 2 X X

3 2 1 0 X 2 2 1 X

5 2 X 2 1 0 2 1 1

6 2 1 2 1 2 0 1 X

n2 c2

0 + 3 2 2 2 X X X X X 3 NO

0 + 5 2 X X 2 2 X X X 3 NO

0 + 6 2 2 X 2 X 2 X X 4 YES

Figure 3.4 Table for Finding 2-Cubes Covering Point o in Example 1

73

13

c0 C1 c2

/ \

\ /

Figure 3.5 Matrix L- of Example 1

74

0 2 3 4 5 6 7 13

3 2 1 0 X 2 2 1 X

5 2 X 2 1 0 2 1 1

6 2 1 2 1 2 0 1 X

n2 C 2

3 + 5 X X 2 X 2 X 2 X 3 NO

3 + 6 X 2 2 X X 2 2 X 4 YES

Figure 3.6 Table for Finding 2-Cubes Covering Point 3 in Example 1

75

13

\

\

d

d

d

0

/

Figure 3.7 Matrix L2 of Example 1

0 2 3 4 5 6 7 13

13 X X X X 1 X X 0

5 X X X 1 0 X 1 1

N 1 C 1

13 + 5 X X X X 1 X X 1 2 YES

Figure 3.8 Table for Finding 1-Cubes Covering Point 13in Example 1

77

C 0 C 1

\

13

\

Figure 3.9 Matrix of Example 1

78

00 01 11 10

01

11

10

Example 3.1Q Karnaugh Map for Example 1

Remarks:

Example 2:

Step 1

Step 2

Step 3

Step 4:

Comparing with the Karnaugh map method in Fig1 0 , the result is obtained algebraically.The so-called "less than cube" (4,5,6 ,7) was not generated.Branching :

F = (0, 2, 3, 4, 5, 7) .: After obtaining matrices A, B, and L (Figs.

11, 12, and 13), it is observed that none of the points are possible core points, and that each is multiple 1-coverable. Starting arbitrarily with point 0 , <0 ,2 >, and <0,4> are found. Points 2 and 4 are the next nodes and L is updated to Lj (Figs. 14 and 15).

: From point 2, point 3 is the next node with<2, 3> the connecting cube. From 4, point 5is the next node with<4,5> the connecting cube. L- is updated to (Figs. 16, 17 and 18).

: Point 3 goes to point 7 by cube <3,7 > andpoint 5 goes to point 7 by cube <5,7>. Thenext node, point 7, in both cases, is a terminal node (Figs. 19, 20 and 21).The branching graph for the connected j- coverable points and the cube-graph corres­ponding to it are developed in Fig. 22 (a)

80

/ \

2 3 2 1 0 1

3 2 1 2 1 0

\ /

Figure 3.11 Matrix A of Example 2

81

p0 P1 p2 p3

/ \

1 2 2 1

\ /

Figure 3.12 Matrix B of Example 2

Figure 3.13 Matrix L of Example

83

0 2 3 4 5 7

0 0 1 X 1 X X

2 1 0 1 X X X

4 1 X X 0 1 X

N 2 C 2

0 + 2 1 1 X X X X 2 YES

0 + 4 1 X X 1 X X 2 YES

Figure 3.14 Table for Finding 1-Cubes Covering Point 0 in Example 2

84

c 0 C 1

/ \

n

n

\

Figure 3.15 Matrix of Example 2

2 1 0 1 X X X

3 X 1 0 X X 1

Ni Cx

2 + 3 X 1 1 X X X 2 YES

Figure 3.16 Table for Finding 1-Cubes Covering Pointof Example 2

86

0 2 3 4 5 7

4 1 X X 0 1 X

5 X X X 1 0 1

Ni CX

4 + 5 X X X 1 1 X 2 YES

Figure 3.17 Table for Finding 1-Cubes Covering Point 4of Example 2

87

/ \

d

\ /

Figure 3.18 Matrix L2 of Example 2

88

0 2 3 4 5 7

5 X X X 1 0 1

7 X X 1 X 1 0

Ni CX

5 + 7 X X X X 1 1 2 YES

0 2 3 4 5 7

3 X 1 0 X X 1

7 X X 1 X 1 0

Ni C1

3 + 7 X X 1 X X 1 2 YES

Figure 3.19 Table for Finding 1-Cubes Covering Points 3 and 5 of Example 2

89

/

\

\

/

Figure 3.20 Matrix of Example 2

90

00 01 11 10

A

Figure 3.21 Karnaugh Map of Example 2

(b)

(c)Figure 3.2 2 Branching Graph and Cube Graph of Example 2

and 22 (b) respectively. The minimum set includes every other branch in the branch­ing graph (Fig. 22 (c) ). (Theorem 7, Corollary 1).

Example 3: Covering of simple branching graph:f(x) = Z(0, 2, 5, 6 , 7,8,10,12,13,14,15)

The two equivalent branching graphs for k=2 are shown in figures 23(a) and (b). Each of these will give the same answer. However, figure 23(a) is preferred be­cause it has two terminal nodes of the first kind. From figure 23(a), it is clear that branches A and D are essen­tial in any minimal set of non-redundant maximal cubes.As for the loop consisting of branches B, C, E, and F, either B and E, or C and F can be selected. If the branching graph shown in figure 23(b) is used, then there are two loops. For the loop on the left side which con­sists of branches A, B and F, A is chosen first. For the loop on the right side which consists of branches C, D and E, D is chosen first. Because node 14 has four branches connected to it while nodes 2,8,7 and 13 each have two branches connected to it, therefore either B or F in the left side loop is chosen. The last branch needed to com­plete the covering the loop on the right side depends on the previous selection of B or F. If B were chosen, E is required. If F were chosen, C is required. There is

93

(7.13)( 7 . 1 4 )

( 2 . 1 4 ) ( 1 3 . 1 4 )

18,.14)

K = 2

( a )

(10.12) ( 1 2 . 1 5 )

(b)

10.00 01 11

00

01

11 E 1 3

10(c)

Figure

3.23

Bran

chin

g Graphs

and

Karnaugh

Map

for

Exam

ple

94

more chance for making erroneous selections in using the branching graph of figure 23(b). This does not pose any difficulty because if a core point is used as a starting node, a branching graph similar to that in figure 23(a) will result. The solution using figure 23(b) illustrates the equivalency between branching graphs. Figure 23(c) is the Karnaugh map for Example 3.

Example 4: Branching graph of different dimensions withinseparable linkage:

f (x) = E (0, 1,2,5,6,7,8,10,12,13,14,15)The branching graph for 2-coverable points is the

same as in Example 3. In Example 4, an additional branch­ing graph for 1-coverable points is shown in Ficrure 24(a). Figure 24(b) depicts the linking of the branching graph for the 1 -coverables to the branching graph for the 2 - coverables. The thick lines indicate the chosen branches, i.e., branches A, C, E, and H. Branches C and E are chosen for the circuit B, C, E, and F. Applying Definition 15 to eliminate branch D, A is essential. Therefore we find B and F are not necessary for the circuit of B, C, E, and F. (Figure 24(c) is the Karnaugh map for Example 4).

Example 5: Branching graph covering with linking fordifferent k but separable:f (x) = (0,1,4.5,8,9,12, 13,16,17,18,19,20,21,

95

22,23,24,25,26,27,28,29,30,31,32,33,36,37, 40,41,43,4 4,45,50,51,52,53,54,55,58,59,60, 61,62,63)

The branching graphs for k = 4 and k = 1 are shown in Figures 25(a) and (b). Although the branching graph is linkable, the linkage of the branching graph does not meet the requirement of Definition 15. Therefore, they are separate branching graphs. (Figure 26 is the linked branching graph. Figure 27 is the Karnaugh map for Example 5. Figure 28 is one of the equivalent branching graphs for 4-coverables in Example 5.)

Example 6 : Having unspecified minterms:

f (x) = £(2,10,12) + Ed (0,1,3,5,7,8,9,11,14,15)

Matrices A, B, and L are shown in Figures 29(a), (b) and (c) respectively. Since only the points in the on-set (2 ,1 0 ,1 2 ) are necessary, these representing don't cares do not appear in matrices A, B and L. These matrices are therefore rectangular. From Figures 29(d) and 29(g), cubes <2, 9> and <12,10> were found, and Figure 29(g) indicates that all on-set points are covered.

The differences in having don't cares are: (1) Therows representing don't cares do not appear in matrices

(a)

96

K = 1

10

( b )

(c)

00 0100

artfg

-GtnGrtfGGrtf«

TtfGrtftflJGaitf ^GO o

i—itn aG E

•G rtf.G XO MGrtf GG 0m M-r

rsj•

m

CL)GGtn

•GIG

98

61

59

Figure 3.26 Branching Graph (Jointed and Covered) for Example 5

000

001

Oil

I 010

110

111

1011

100

99

A

41

61

54 5551

2122 20

313025 28

100101'010 111000 Oil 110

Figure 3.27 Karnaugh Map for Example 5

100

0D

T o 2 0 , s a m e a s

< 1 8 , 2 9 > = E

Figure 3.2 8

T o 6 3 , s a m e a s

< 2 7 , 5 4 > = D

E

< B >

T o 9 , s a m e a s

< 0 , 4 5 > = A

T o 2 0 , s a m e a s

< 2 9 , 3 6 > = B

Branching Graph for Example 5 (Different but Isomorphic to Figure 3.25)

101

2 10 12 0 1 3 5 7 8 9 11 14 15/

10

12

\0 1 3 1 2 1 3 2 2 3 2 2 3

1 0 2 2 3 2 4 3 1 2 1 1 2

3 2 0 2 3 4 2 3 1 2 3 1 2\

(a)

p0 P1 p2 p 3 p 4/ \

1 3 5 4 0

10 1 4 5 2 1

12 1 2 5 4 1\ /

(b)

C0 C1 C2 C3 C4

10

12

\c 0

c 0 0

(c)

Figure 3.29 Diagrams for Example 6

(a) Matrix A(b) Matrix B(c) Matrix L

102

2 10 12 0 1 3 5 7 8 9 11 14 15

2 0 1 3 1 2 1 3 2 2 3 2 2 3

1 2 3 2 0 2 3 X 2 3 1 2 3 1 2

5 3 X 2 2 1 2 0 1 3 2 3 3 2

9 3 2 2 2 1 2 2 3 1 0 1 3 2

15 3 2 2 X 3 2 2 1 3 2 1 1 0

n3 C3

2 + 12 3 3 3 3 X X X X 3 X X 3 X 6 NO

2 + 5 3 X X 3 3 3 3 3 X X X X X 6 NO

2 + 9 3 3 X 3 3 3 X X 3 3 3 X X 8 YES

2 + 15 3 3 X X X 3 X 3 X X 3 3 3 7 NO(d)

Figure 3.29 (cont.)(d) Table for Finding

3-Cubes(e) Matrix Lj

21012

Cq Ci C2 C3/ \- - e

dc 0/

(e)

103

2 10 12 0 1 3 5 7 8 9 11 14 15

1 2 X 2 0 2 X X 2 X 1 2 X 1 2

1 0 X 0 2 2 X X X X 1 2 X 1 2

0 X 2 2 0 X X 2 X 1 2 X X X

5 X X 2 2 X X 0 X X 2 X X 2

9 X 2 2 2 X X 2 X 1 0 X X 2

15 X 2 2 X X X 2 X X 2 X 1 0

(Ddet•H+>cou

CTiC N

•m0)H3tn■rH

N2 C2

12 + 10 X 2 2 X X X X X 2 X X 2 X 4 YES

1 2 + 0 X X 2 2 X X X X 2 X X X X 3 NO

1 2 + 5 X X 2 X X X 2 X X X X X X 2 NO

1 2 + 9 X X 2 X X X X X 2 2 X X X 3 NO

12 + 15 X X 2 X X X X X X X X 2 2 3 NO

(f) Table for Finding 2-.Cubes

/

10

12\

e

\e

- - - d

/

C N(dX•Hu+>5tn

1 0b

ii0100

12

Ad9

d5

dlld15d 7

14

A 100 01 11 10

CX<daX!tnd(CCnm«

(g)

104

A, B and L; (2) The distance row has to be established, only when the don't care point is a potential candidate for a diametrical extreme of a cube that covers an on-set point. Even then, only the relevant entries appear in the row. For example, in Figure 29(d), only the rows re­presenting don't care points 5, 9 and 15 are needed. In Figure 29(f), although the don't care points 0,5,9 and 15 are needed, the entries of these rows under the columns of 2,1,3,7 and 11 are irrelevant. In this case, we are considering a 2 -cube covering for point 1 2 , yet all the distances from the points (2,1,3,7 and 11 to 12) are greater than 2. Therefore, a 3-cube covering cannot be formed. In the case where the majority of points are don't cares, then a covering for the on-set and the off­set would each be found and then combined by Nelson's method (N3 ) or McCluskey's method ( Me2) .

3.3 CONCLUSIONThe concept of distance matrices and distance-distri-

bution matrices leads to the algorithm of finding the maximal cube covering starting with the largest possible cube first. This is the reverse procedure to those using conventional tabular methods. Obviously, when larger cubes exist, it is thus more effective. However, when there are numerous small cubes, the algorithm tends to increase the labor while the conventional methods tend to decrease it.

105

Initial work has not been effective in ascertaining the break even point. Presently, computer programming to minimize the effort is being studied.

Although similar in some respects to the map method, this method is more powerful and effective. Firstly, it does not depend upon the human faculty of recognizing geometric patterns; and secondly, it is systematic and precise and. can be readily computerized.

The concept of the branching graph not only leads to solving the PI table covering problem more effectively, but also gives a definite way of finding all the maximal cubes.

Since the new approach breaks down the on-set of a switching function into connected j-coverable subsets, the maximal cube table (or prime-implicants table) evolves into an easier soluction. In fact, the branching graph shows more interrelationships between the maximal cubes and the elements of the on-set of a switching function than the row dominance and the column dominance in a prime-implicant table. Such relations include the dia­metrical opposite points of a cube, the elimination of less-than cubes, the true non-essential maximal cubes, and the connectivity of the maximal cubes.

Because of the corresponding characteristics between the Karnaugh map and the new approach, it is conceivable

106

that many problems in switching theory and logic design can be solved with the new approach. These may include the multi-output problems, the switching functions with feedback and sequential digital circuits. The full im­pact of this method cannot be realized until more types of problems are studied.

CHAPTER IV THE APPLICATION OF GEOMETRIC APPROACH

4.1 Algorithm for two level Logic Minimization.Following the analysis in Chapter 2 Section 4, we

can define the characteristics function for an array of the on-set and don't cares of a switching function as following:Characteristic 1: The highest coverable dimension.Characteristic 2: Core point candidacy.Characteristic 3: Probability of sharing common parts.Characteristic 4: The accumulative awayness.The general algorithms as follows:MAIN PROGRAM(1) Read in number of variables and number of don't care

minterms, minterm zero is included or not, data card of all minterms in the on-set and don't care set.

(2) Calculate the maximal cube dimension and binomial coefficients for that dimension.

(3) Calculate the distance Matrix A for all the minterms(4) Write matrix A into disc file.(5) Calculate the distance frequency distribution for

each minterm. Then immediately calculate Char. 1 and Char. 2.

(6 ) If d (I,X) > 1 for all X in the array, I is isolated(7) Print out all isolated minterms.

10 8

(8) Char (1,1) = Characteristic 1 of minterm I = thehighest coverable dimension for satisfying the follow­ing two criteria.

(i)k=j

= Min k=o

bik&

Char(1,1) (Theorem 3)

(ii) The number of minterms in the array meeting1 . .condition 1 is more than 2 1 3 .

(9) If Frequency of distance = 1 or 2^ where j is the highest coverable cube covering, then the minterm in examination is a candidate of core point. Char. (1,2) = 1 or 2^, otherwise Char. (1.2) = 5 or 999999, which means multiple covering.

(10) Check if Maximal Cube dimension is valid or not.If not, lower Maximal-Cube dimension by 1. The criterion is l c ^ (number of maximal cube coverable minterms) _> maximal cube dimension.

(11) Characteristic 3 = Char (1,3) = 0 for on-set= 999999 for don't care,

at the start of the program. It will change as the process of finding cube covering progresses. At the start the on-set are not common with any cube cover­ing and the don11 care minterms are in common with every cube covering. Therefore the probability of sharing any common part for the on-set minterms is

109

zero, for don't care minterms is 1 , for which we usef■999999/10 to approximate and then conveniently drop

/rthe common denominator, 1 0 .(12) Test all the core point candidates as to whether

there is a cube or not. If the cube exists, then this minterm is covered and all the minterms in this cube are also covered. Therefore their Characteristic 3 all become 99999. The probability of sharing is certain for future cube covering. If not, then this minterm being examined becomes a j- 1 coverable min­term.After a core point and core cube is established, the accumulative awayness is computed for the rest of the minterms in the array.

(13) Use the criteria that a new cube should have the least awayness from and the least intersection with the don't care set or covered set. The rest of the array is gradually covered with the minimum maximal- cube covering for the whole array.

SUBROUTINE CORE CUBES(1) Pick any minterm that is a core point candidate; if

all possible candiates have been checked stop.(2) Check characteristic 3 to single out don't cares.(3) Use characteristic 1 for dimension. Call subroutine

CUBE to verify the existence of the core cube.

110

(4) If core point candidate does not have a cube cover­ing, lower the dimension and charge the char (3 .2 ) to non core candidate the minterm being processed.

(5) If indeed the cube exists, then print out the data for this cube.

(6 ) If not finished, return to step 1.

SUBROUTINE MULTIPLE COVERABLE POINTS.1. Select starting point.

a. Highest coverable by Char (1,1)b. Smallest awaynessc. Largest aloofness.d. Least probability to share common parts with

existing maximal-cube covering.2. Call subrountine cubes to verify the existance of

all the highest maximal cubes.3. Check the actual number of common parts between the

cubes found in 2 . and choose the one that has thefewest common parts.

4. Checking less-than-cubes by characteristic 1 to check actual coverable dimension.

5. Starting dimension 1 to dimension maximal.6 . Go through 1 through 3.

SUBROUTINE CUBES1. Called by main program minterm = MTH, Dimension = D,

Type = ZNTH.

Ill

2. Read from file the distance function for Mth.3. If ZNTH = 1, core point candidate verify the existing

of a cube and print out otherwise, go to 5.4. Calculate accumulated awayness, relative probability

of showing common parts go to 7.5. Calculate probability of share verigy all D-cubes

covering MTH. If awayness are the same, then compare the probability of share.

6 . Calculate accumulative awayness and aloofness.7. Return to main program. The detailed flow diagrams

are shown in Figure 4.1 for the main program. Figure4.2 shows the sub-routine of finding the core cubes. 4.3 is program for finding the remainder of the minimum maximal cube covering, and Figure 4.3 shows the sub­routine cube identifying or cube. The listing ofthe Fortran IV programs is given in the appendix. A detailed example is worked out in the next section to demonstrate the mechanism of the program.

112I N P U T N V , N D C C A L C U L A T E

M I N T E R M S 1M A X C U B < L o g ?

( # O F M I N T E R M S ) 2

# M A X C U B - C O V

2 * * m a x c u b ?

C A L C U L A T E

D I S T A N C E 3 M A T R I X A

F I L E 4

M A T R I X A

C A L C U L A T E

D I S T A N C E F R E Q U E N C Y

M A T R I X B 5

d ( I , X ) = 6 N

F O R N O X 7

N O a ^ ) ------- ---- ----- ---- ----- ----

fC H A R ( 1 , 1 ) = J

I : j - C O V E R A B L E

C H A R ( 1 , 2 ) = i

I : C O R E P O I N T

C A N D I D A T E

C H A R ( 1 , 2 ) = 5

I : M U L T I - C O V 8

Y E S

N O

Y E S ^r

C H A R ( I , 3 ) = 0I : C A R E

C H A R ( 1 , 3 ) = 9 9 9 9

I : D O N ' T C A R E 11

I : I S O L A T E D

7 M I N T E R M

M A X C U B

m a x c u b - 1 10

I

N O

A L L C O R E P O I N T

C A N D I D A T E S T E S T E D

^ S T O P 1 6 ^

Y E S

A L L C O V E R E D

: >N O

C A L L S U B R O U T I N E

S E A R C H F O R A L L 1 3

V tC A L L S U B R O U T I N E 1 4

S E A R C H F O R N O N - C O R EE S S E N T I A L C O R E

M A X I M A L C U B E S

( f i g - 4 . 2 p a g e 1 1 3 )

F I G . 4 . 1 F L O W C H A R T

O F M A I N P R O G R A M

M A X I M A L C U B E S

( f i g . 4 . 3 p a g e 1 1 4 )

113

A L L C O R E M A X I M A L

C U B E S H A V E B E E N

S E A R C H E D

N O1 6 < N T

1 6 = 1 6 + 1

Y E S

Y E S C H A R ( I , 3 ) = 9 9 9 9

( D O N ' T C A R E S ) ?

N O

N OC H A R ( 1 , 2 ) = 1 ? v

( c o r e p o i n t c a n d i d a t e s ) 4

Y E S

D = C H A R ( 1 , 1 )

( D I M E N S I O N )

M T H = 1 6

( E N D P O I N T )

C A L L S U B R O U T I N E ( F i g . 4 > 4PaSe 115) 6C U B E ( D , M T H )

C H A R ( 1 , 1 )

= C H A R ( I , 1 ) - 1

( l o w e r j - c o v e r a b l e

d i m e n s i o n )

C H A R ( I , 2 ) = 5

( n o t a c o r e p o i n t )

N OC U B E E X I S T S ?

Y E S

P R I N T O U T

T H E C U B E

F I G U R E 4 . 2 F l o w C h a r t o f S u b r o u t i n e S e a r c h f o r A l l E s s e n t i a l C o r e

M a x i m a l C u b e s ( c a l l e d b y m a i n p r o g r a m i n F i g u r e 4 2 B l o c k 1 3 )

114

D O 1 = 1 , M A X C U B , 1 L f J L / ^ p ' N

I M A X C U B ? 1 8 7

< 1 Y E S

D O J 2 = 1 , N T , 1 JL D O J = 1 , N T 1

J 2 = N T ? 2 N O J 6= N T 1 7

N 0 ^CHARCje^) 1fi YES 2 * * 1 ?

Y E S

i \ y —. ■ ■T y e s / : h a r ( j 6 . 3 )

~ J \ p H A R ( J 6 , l )

N O

Y E S

P S H A R E = C H A R ( J 2 , 3 )D = D - 1C L O S E = C H A R ( J 2 , 4 )

M T H - J 2 ; D - I 3 M T H = X

O R D E R - C H A R ( J 2 , 2 )Z N T H = 5

N O

1 4 <C U B E

E X I S T S 1 3

T

D O 1 6 = 1 , N T , 1

I 6= N T ?

N O M T H = X

D = I 1 1 Z N T H = 5

Y E S / 1 6 = J 2 ?

/ C H A R ( I 6 , 1 ) = 1 ?

C H A R ( I 6 , 2 ) = 2 * * i ? 5

\ C H A R ( I 6 , 3 ) = P S H A R E ?

N O

N O£

C H A R ( I 6 , 3 )

V=P S H A R E ?

Y E S

<

C A L L S U B R O U T I N E

C U B E ( X . D . S ) 1 2

( f i g . 4 , 4 p a g e 1 1 5

C H A R ( I 6 , 4 )

C L O S E ?

\ n o I n o /N 0 ,____/ C H A R ( I 6 , 4 ) \ Y E S

C L O S E ? J l

XP S H A R E = C H A R ( 1 6 , 3 )

M T H = 1 6 1 0

IN l / C H A R ( 1 6 , 2 ) \ Y E ^

\ ORDER ? /

F I G U R E 4 . 3 S U B R O U T I N E S E A R C H I N G F O R N O N - C O R E M A X I M A L C U B E S

( C a l l e d b y m a i n p r o g r a m f i g . 4 . 4 b l o c k 1 4 , p a g e 1 1 2 )

115( C A L L E D

D I A N = 0

D = D - l 13

M T H , D

Z N T H Y E S __ _ _ _ _ _ _ _ _ _ _ _

/ N O C U B E

\ F O U N D ?

C U B E ( M T H , S ) P R I N T C U B E

R E A D F I L E F O R T H E

D I S T A N C E F U N C T I O N O F M T H I N T O T 1 ( I )

R E P L A C E ( M T H , I ) A S

T H E N E W ( M T H , S )

N O^ D O 1 = 1 , N T

F O R T 1 ( I ) = DD O E S ( M T H , I ) H A V E

M O R E N E W M I N T E R M S

T H A N ( M T H , S ) ?

N O E S

Y E S

N O

C U B E ( M T H , I )

E X I S T S ?N O

L E T ( M T H , I )

= ( M T H , S )

Y E S

N O ( W A S ( M T H , S ) F O U N D ~ ?Z N T H =

Y E S

Y E SN O C O R E M A X I M A L C U B E ( M T H , I )

F O U N D A N D P R I N T O U T

Z N T H =

E R R O R D I A N = 1 a n d 1 a n d c a l c u l a t e

P S H A R E = D I F C H A R ( X , 1 ) ~ D + W a b

C H A R ( X , 1 ) - W a b I F C H A R ( X , 1 ) - D + W .t o F i g . 4 . 2 t > l o c k

o r

t o F i g ^ . j b l o c l c

10R E T U R N

F I G U R E 4 . 4 F L O W C H A R T O F S U B R O U T I N E C U B E

( C a l l e d b y F i g ^ ^ b l o c k 6 , o r C a l l e d b y F i g ^ ^ b l o c k 1 2 )

116

4.2 Computer format and mechanism illustration. The computer program implemented in the Univac VMOS SYSTEM at New Jersey Institute of Technology can be illustrated as following:

1) Step 1. enteringthe number of don't cares, and the number of variables.Ihen enter all the minterms (on-set plus don't cares with don't cares after on-set as follow:

* 5 5 5* 1 2 3 4 5 6 7 9 11 12 13*14 15 17 2 0 23 24 25 27 28*g

Step 2 : The computer will calculate the distancefunction and store them in a disc file. Then cal-culate all the charateristic and point out

MAXIMUM CUBE DIMENSION = 41 0 0 0 2 1 0 0 3 3 1 0 4

6 4 1

THE CHARACTERISTICS:4 1 2 1 4 1 4 1 4 1 4 1 45 4 1 4 1 4 1 4 1 3 1 4 12 1 2 1 1 1 2 1 3 5 2 1 31

Step 3, the computer will start to search all the

core cubes first; and point out, the relative Drobability of sharing (checking points) , the accumulated awayness, the cube found and cube selected. When all the core cubes has been reached, it will then Doint out a statement of the fact and the maximum cube dimension of all the core cubes. A sample is as following:CHECKING POINTS

2 9999 9999 22 0 0 9999

ACCUMULATED AWAYNESS:2 0 02 4 4

CUBE ( 2 . 7 CUBE ( 14 . 5CHECKING POINTS

4 9999 9999 99999999 0 1 9999

ACCUMULATED AWAYNESS:4 0 02 8 6

CUBE ( 14 , 5CUBE ( 17 , 9CHECKING POINTS9999 9999 9999 99999999 9999 1 9999

2 9999 9999 09999 9999 9999 9999

2 2 0 0 40 0 0 0 0

) TS SELECTED.) HAS BEEN FOUND.

9999 9999 9999 29999 9999 9999 9999

2 2 0 0 60 0 0 0 0) IS SELECTED) HAS BEEN FOUND,

9999 9999 9999 99999999 9999 9999 9999

2 0 0 2

2 4 4 2

, 4 9999 -9999 9999

4 4 4 2

6 9999 9999 9999

118

ACCUMULATED AWAYNESS :4 0 0 2 2 0 0 6 6 4 4 2 2 8 10 0. 0 Q 0 0

CUBE ( 1 7 , 9 ) IS SELECTED.CUBE ( 20 , 1 2 ) HAS BEEN FOUND.

CHECKING POINTS9999 9999 9999 9999 9999 9999 9999 9999 6 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999

ACCUMULATED AWAYNESS:4 0 0 2 2 0 0 6 12 4 42 2 8 10 0 0 0 0 0

CUBE ( 2 0 , 1 2 ) IS SELECTEDALL CORE CUBES HAVE BEEN SEARCHED,AND THE MAXIMUM CUBE DIMENSION IS: 3

Step 4: after all the core cubes has been found, then itstart to look for the rest of the minimum-set of maximal cubecovering and prints out like this:CUBE ( 11 , 5 ) HAS BEEN POUNDCHECKING POINTS

9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999

ACCUMULATED AWAYNESS:4 0 0 2 2 0 0 6 12 4 42 2 8 1 0 0 0 0 0 0

CUBE ( 1 1 , 5 ) IS SELECTED**PORTRAN ** STOPNotice the computer also prints'out two up-dated value of charac­teristic. CHECKING POINTS abb the relative probability of sharing. Accumulated Awayness is also printed out. This is for de-bugging and applications other than just the minimum set. For minimization only all this can be omitted. At this point, all the calculation is completed. And the results is output as follow:Total Number of On-set and Dont*Cares = 20ON-SET = 15 DONT-CAKES = 5NUMBERS OF VARIABLES = 5

120

MINT.# 1 = 1 MINT.# 2 = 2 MINT.# 3 =MINT.# 4 = 4 MINT.# 5 = 5 MINT.# 6 =MINT.# 7 = 7 MINT.# 8 = 9 MINT.# 9 =MINT.# 1 0 = 1 2 MINT.# 11 = 13 MINT.# 1 2 =MINT.# 13 = 15 MINT.# 14 = 17 MINT.# 15 =MINT.# 16 = 23 MINT.# 17 = 24 MINT.# 18 =MINT.# 19 = 27 MINT.# 20 = 28 MINT.#

THE CUBE COVERING LIST:The 1-TH CUBE COVERING IS:

2-CUBE = ( 2 ,1. ) = 00-1-THIS CUBE COVERS THESE MINTERMS J

2 7 3 6

THIS MINTERM MT ( 2 ) = 2 IS A CORE POINTTHE 2-TH CUBE COVERING IS:

3-CUBE = ( 14 , 5 ) = 0-1—THIS CUBE COVERS THESE MINTERMS:

14 5 4 6 7 12 13 15THIS MINTERM MT( 12 ) = 14 IS A CORE POINTTHE 3-TH CUBE COVERING IS:

2-CUEE = ( 17. , 9 ) = — 001THIS CUBE COVERS THESE MINTERMS:

17 9 1 25THIS MINTERM MP( 14 J = 17 IS A CORE POINTTHE 4-TH CUBE COVERING IS:

2-CUBE = ( 20 , 12 ) = — 100THIS CUBE COVERS THESE MINTERMS:

20 12 4 28THIS MINTERM MT ( 15 ) = 20 IS A CORE POINTTHE 5-TH CUBE COVERING IS:

3-CUBE = ( 11 , 5 ) = 0---1THIS CUBE COVERS THESE MINTERMS:

11 5 1 3 7 9 13 15

121

THE FINAL CHECKING205 101 205 205 205 205 205 205 105 205 205 101205 101 101 109 109 209 109 209

The final checking indicates what status of the maximal cubecoverings and also checks whether all on-set is covered .A second completed illustrative example is reproduced asfollowing to show other features.In the appendices, a collection of various example problems has been selected with the out-put result print out. For reference purpose and better understanding of the mechanics of the computer program. Those checking arid debug out-put print out are also en­closed following seme of the examples results out-put print. Illustrative Examples 2.FORTRAN IV PROGRAM LARGEE STARTED--6 9

0 1 2 5 8

.13 18 19 28 34 37 39 47 56 574 12 16 20 23 38 49 59 629

MAXIMUM CUBE DIMENSION = 41 0 0 0 2 1 0 0 3 3 1 0 4 64 1

THE CHARACTERISTICS:3 5 2 1 3 1 3 5 2 1 2 1 3 1 2 1 2 1 2 1 2 1 3 1 1 1 1 1 1 - 5 1 1 4 1 3 5 3 1 3 1 1 1 2 1 1 1 1 1 0 0

MAXIMUM CUBE DIMENSION HAS BEEN LOWERED TO: 3CUBE ( 1 , 4 ) HAS BEEN POUND.

CHECKING POINTS9999 9999

0 0 9999

ACCUMULATED AWAYNESS:

2 9999 1 1 0 0 0 0 1 00 0 9999 9999 9999 9999 9999 9999 9999 9999

060

06

26

CUBECUBE

1

844

CHECKING POINTS9999 9999

0 0 9999

0120

010

CUBE ( CUBE (

813

CHECKING POINTS9999 9999

0 0 9999

ACCUMULATED AWAYNESS:0160

016

06

20

20

40

40

40

40

20

40

) IS SELECTED.) HAS BEEN POUND.

3 9999 9999. . 2 0 0 1 0 1 00 0 9999 9999 9999 9999 9999 9999 9999 9999

ACCUMULATED AWAYNESS:412

44

012

20

40

80

100

60

80

60

100

) IS SELECTED.) HAS BEEN POUND.

3 9999 9999 9999 0 0 2 0 2 00 0 9999 9999 9999 9999 9999 9999 9999 9999

818

016

20

40

14.'0

16 . 0

80

140

80

140

CUBE ( 13 , 4 ) IS SELECTED.MAXIMUM CUBE DIMENSION HAS BEEN LOWERED TO:CUBE ( 28 , 4 . ) HAS BEEN POUND

123

CHECKING POINTS 9999 9999

0 0

3 9999 9999 9999 0 0 9999 0 20 0 0 9999 9999 9999 9999 9999 9999

9999 9999 9999ACCUMULATED AWAYNESS

0 0 1 2

2 0 2 2 2 0

0 0 0

CUBE ( 28 , 4CUBE ( 47 , 39CHECKING POINTS

9999 9999 39999 9999 0

9999 9999 9999ACCUMULATED AWAYNESS:

0 0 182 0 2 2 280 0 0

CUBE ( 47 , 39

0

242

2040

180

220

80

200

120

IS SELECTED.HAS BEEN FOUND.

3 9999 9999 9999 0 99990 9999 9999 9999 9999 9999 9999

0

302

2440

260

280

80

240

140

CUBE 56 57 )IS SELECTED.HAS BEEN FOUND.

CHECKING POINTS 9999 9999 0 99993 9999 9999 9999 09999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999

ACCUMULATED AWAYNESS:

124

0 Q 36 Q 2 4 40 40 8 382 0 2 2 28 30 26 0 0 0 0 0

0 0 0

CHECKING POINTS9999 9999 3 9999 9999 9999 0 0 9999 0 29999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999

ACCUMULATED AWAYNESS:0 0 36 0 2 4 40 40 8 38

2 0 2 2 28 30 26 0 0 0 0 0

0 0 0

CUBE ( 61 , 57 ) IS SELECTED.ALL CORE CUBES HAVE BEEN SEARCHED,AND THE MAXIMUM CUBE DIMENSION IS : 2CUBE ( 18 , 0 ) HAS BEEN FOUND.

125

CHECKING POINTS9999 9999 9999 9999 9999 9999 9999 09999 9999 9999 9999 9999 9999 9999 9999 9999

ACCUMULATED AWAYNESS:

9999 0 2 99999999 9999 9999 9999

0 0 36 0 2 4 40 42 8 40 82 2 28 30 26 0 0 0 0 0 0 0

30 2 00 00

400

CUBECUBECUBE

( 18 ( 34( 34

0

238

WATCH FOR 2 CHECKING POINTS

AND

)))

38

IS SELECTED HAS BEEN FOUND. HAS BEEN POUND.

9999 9999 9999 9999 9999 9999 9999

99999999

9999 9999 9999 9999 9999 9999

09999

99999999

99999999

29999

ACCUMULATED AWAYNESS:0 0 36 22 28 30

026.

2 4 40 0 0 0

480

80

400

400

CUBE ( 1 9 , 2 3 ) IS SELECTEDCUBE ( 37 , 5 ) HAS BEEN POUND.CUBE ( 3 7 , 3 9 ) HAS BEEN FOUND.CHECKING POINTS

9999 9999 9999 9999 9999 9999 9999

99999999

9999 9999 9999 9999 9999 9999

99999999

99999999

99999999

99999999

ACCUMULATED AWAYNESS:0 0 36 22 28 30

0 2 4 40 266 0 0 0

480

80

400

400

200

200

126

CUBE (_ 37 , 5 ) IS SELECTED.**FOKTRAN ** STOP

TOTAL NUMBER OF ON-SET AND DQNT*CARES = 25ON-SET = 16 DQNT-CAKES = 9NUMBERS OF VARIABLES - 6

MINT.# 1 = .0 MINT.# 2 = 1 MINT.# 3 = 2MINT.# 4 = '5 MINT.# 5 = 8 MINT.# 6 = 13MINT.# 7 = 18 MINT.# 8 = 19 MINT.# 9 = 28MINT.# 1 0 = 34 MINT.# 1 1 = 37 MINT.# 1 2 = 39MINT.# 13 = 47 MINT.# 14 = 56 MINT.# IS = 57MINT.# 16 = 61 MINT.# 17 = 4 MINT.# 18 = 1 2MINT.# 19 = 16 MINT.# 2 0 = 2 0 MINT.# 2 1 = 23MINT.# 2 2 = 38 MINT.# 23 = 49 MINT.# 24 = 59MINT.# 25 = 62 MINT.#

MINTERM # 25 = 62 IS ISOLATED AND ESSENTIAL

127

THE CUBE COVERING LIST:

THE 1-TH CUBE CORERING IS : 2-CUBE = ( 1 , 4 ) =

THIS CUBE COVERS THESE MINTERMS 1 4 0 5

THIS MINTERM MT ( 2 ) = 1THE 2-TH CUBE COVERING IS :

2-CUBE = ( 8 , 4 ) -THIS CUBE COVERS THESE MINTERMS

8 4 0 12THIS MINTERM MT( 5 ) = 8

THE 3-TH CUBE COVERING IS : 2-CUBE = ( 13 , 4 ) =

THIS CUBE COVERS THESE MINTERMS 13 4 5 12

THIS MINTERM MP ( 6 ) = 13

THE 4-TH CUBE COVERING IS : 2-CUBE = ( 28 , 4 ) =

THIS CUBE COVERS THESE MINTERMS 28 4 12 20

THIS MINTERM MT ( 9 ) = 2 8THE 5-TH CUBE COVERING IS :

1-CUBE = ( 47 , 39 ) =THIS CUBE COVERS THESE MINTERMS

47 39THIS MINTERM MT (. 13 ) - 47THE 6-TH CUBE COVERING IS :

1-CUBE = ( 56 , 57 ) =

0 0 0-0-

IS A CORE POINT

0 0 — 00

IS A CORE POINT

00- 10-

IS A CORE POINT

0— 100

IS A CORE POINT

10-111

IS A CORE POINT

= 11100“

128

THIS CUBE COVERS THESE MINTERMS :56 57

THIS MINTERM MT ( 14 ) = 56 IS A CORE POINT

THE 7-TH CUBE COVERING IS:1-CUBE = ( 61 , 57 ) = 111-01

THIS CUBE COVERS THESE MINTERMS :61 57

THIS MINTERM MT ( 16 ) = 61 IS A CORE POINT

THE 8-TH CUBE COVERING IS :2-CUBE = ( 18. , 0 ) = 0-00-0

THIS CUBE COVERS THESE MINTERMS : 18 0 2 16

THE 9-TH CUBE COVERING IS :1-CUBE = ( 34 , 38 ) - 100-10

THIS CUBE COVERS THESE MINTERMS : 34 38

THE 10-TH CUBE COVERING IS :1-CUBE = ( 19 , 23 ) = 010-11

THIS CUBE COVERS THESE MINTERMS : 19 23

THE 11-TH CUBE COVERING IS :1-CUBE = ( 37 , 5 ) = 00101

THIS CUBE COVERS THESE MINTERMS : 87 5

129

THE FINAL CHECKING105 101 105 105 101 101 105 105 101 105 105101 101 205 101 105 105 105 105 2Q9 209 109109

I) The tracing of cyclic cubes:The computer is capable to trace out the cyclic cubes

as shown in the follwing example.

TOTAL NUMBER OF ON-SET AND DQNT*CARES = ON-SET = 14 DONT=CAKES = 0NUMBERS OF VARIABLES - 5

14

MINT.# 1 = 1 MINT.# 2 = 4 MINT.# 3 = 9MINT.# 4 = 11 MINT.# 5 = 12 MINT.# 6 = 27MINT.# 7 = 26 MINT.# 8 = 29 MINT.# 9 = 28MINT.# 1 0 = 18 MINT.# 1 1 = 2 2 MINT.# 12 = 23MINT.# 13 = 21 MINT.# 14 = 0 MINT.#

105109

THE CUBE COVERING LIST:

THE 1-TH CUBE COVERING IS :1-CUBE = ( 0 , 4 ) = 00-00

THIS CUBE COVERS THESE MINTERMS : 4 0

THE 2-TH CUBE COVERING IS :1-CUBE = ( 12 , 28 ) = — 1100

THIS CUBE COVERS THESE MINTERMS : 1 2 28

THE 3-TH CUBE COVERING IS :1-CUBE = ( 21 , 29 ) = 1-101

THE 4-TH CUBE COVERING IS :1-CUBE = ( 22 , 23 ) = 1011-

THIS CUBE COVERS THESE MINTERMS : 22 23

THE 5-TH CUBE COVERING IS :1-CUBE = ( 18 , 26 ) = 1-010

THIS CUBE COVERS THESE MINTERMS : 26 18

THE 6-TH CUBE COVERING IS :1-CUBE = ( 27 , 11 ) = -1011

THIS CUBE COVERS THESE MINTERMS : 11 27

THE 7-TH CUBE COVERING IS :1-CUBE = ( . 1 , 9 1 = 0-001

THIS CUBE COVERS THESE MINTERMS :1 9

THE FINAL CHECKING105 105 105 105 105 105 105 105 105 105 105 105105 105

The way it trace out of the cyclic cubes is shown in the checking print out of the canputer as follow:

* 5* 1 4 9 11 12 27 26 29 28 18 22 23 21*9

MAXIMUM CUBE DIMENSION = 3

1 0 0 2 1 0 3 3 1

THE CHARACTERISTICS :2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 22 1 2 1 2 1

MAXIMUM CUBE DIMENSION HAS BEEN LOWERED TO: 2MAXIMUM CUBE DIMENSION HAS BEEN LOWERED TO: 1ALL CORE CUBES HAVE BEEN SEARCHED,AND THE MAXIMUM CUBE DIMENSION IS : 1CUBE ( 0 , 1 ) HAS BEEN FOUND.CUBE ( 0 , 4 ) HAS BEEN FOUND.

132

WATCH FOR 1 AND

CHECKING POINTS0 9999 0 00 9999

ACCUMULATED AWAYNESS2 0 4 14 0

CUBE ( 0 f 4CUBE ( 12 / 4CUBE ( 12 28

CHECKING POINTS0 9999 0 00 9999

ACCUMULATED AWAYNESS8 2 8 12 8 4

CUBE ( 1 2 , 2 8CUBE ( 2 1 , 2 9CUBE ( 2 1 , 2 3WATCH FOR 29 AND

CHECKING POINTS0 0 0 0 0 0 0

9999 9999

4

0 0 0 0

2 8 6 6

) IS SELECTED.) HAS BEEN POUND.) HAS BEEN FOUND.

9999 0 0 0

2 14 10 8

) IS SELECTED.) HAS BEEN FOUND.) HAS BEEN FOUND.23

9999 0 0 9999

0 0 0 0

4 4 4 6

9999 0 0 0

4 10 8 12

9999 0 0 0

133

ACCUMULATED AWAYNESS :1 28

61 0

12 18 6 18 16 8 6 16 12 14

CUBE ( 2 1 9 29 ) IS SELECTED.CUBE ( 2 2 9 18 ) HAS BEEN FOUND.CUBE ( 2 2 9 23 ) HAS REEN FOUND.

WATCH FOR 18 AND 23CHECKING POINTS

0 9999 0 0 9999 0 0 9999 9999 0 9999 99999999 9999

ACCUMULATED AWAYNESS :24 16 26 28 18 24 20 18 14 18 14 1816 2 0

CUBE ( 1 8 , 2 6 ) IS SELECTED,iJCUBE ( 2 7 , 1 1 ) HAS BEEN FOUND.CUBE ( 2 7 , 2 6 ) HAS BEEN FOUND.

134

CHECKING POINTS0 9999 0 9999 9999 9999 9999 9999 9999 9999 9999 9999

9999 9999

ACCUMULATED AWAYNESS :28 24 28 28 24 24 22 222 2 26

CUBE (. 27 , 1 1 ) IS SELECTED.CUBE ( 1 , 9 ) HAS BEEN FOUND.CUBE ( 1 , 0 ) HAS BEEN FOUND.

CHECKING POINTS9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999

ACCUMULATED AWAYNESS :28 28 28 30 28 28 28 26 26 28 28 2826 28

CUBE ( 1 , 9 ) IS SELECTED.

135

2) To find out isolated minterms :-40 3 5 6 12 15 9 . 109

MAXIMUM CUEE DIMENSION = 31 0 0 2 1 0 3 3 1

THE CHARACTERISTICS :0 0 0 0 0 0 0 0 0 0 0" 0 0 0 0 0

MAXIMUM CUBE DIMENSION HAS BEEN LOWERED TO: 2MAXIMUM CUBE DIMENSION HAS BEEN LOWERED TO: 1MAXIMUM CUBE DIMENSION HAS BEEN LOWERED TO: 1MAXIMUM CUBE DIMENSION HAS BEEN LOWERED TO: 0ALL CORE CUBES HAVE BEEN SEARCHED,AND THE MAXIMUM CUEE DIMENSION IS : 0*9

**FORTRAN ** STOP

TOTAL NUMBER OF ON-SET AND DQNT*CARES = 8ON-SET = 8 DONT=CARES = 0NUMBERS OF VARIABLES = 4

MINT.# 1 r= 0 MINT.# 2 = 3 MINT.# 3 =MINT.# 4 = 6 MINT.# 5 = 12 MINT.# 6 =MINT.# 7 = 9 MINT.# 8 = 10 MINT.#

MINTERM # 1 = 0 IS ISOLATED AND ESSENTIALMINTERM # 2 = 3 IS ISOLATED AND ESSENTIALMINTERM # 3 = 5 IS ISOLATED AND ESSENTIALMINTERM # 4 = 6 IS ISOLATED AND ESSENTIALMINTERM # 5 — 1 2 IS ISOLATED AND ESSENTIAL

MINTERM # 6 = 15 IS ISOLATED AND ESSENTIALMINTERM # 7 = 9 IS ISOLATED AND ESSENTIALMINTERM # 8 = 10 IS ISOLATED AND ESSENTIAL

137

4.3 CONCLUSION:The canputer programming was done on a relatively slow

and limited storage teaching computer to demonstrate the theory of minimal maximal cube coverings. It is possible to improve the efficiency and enlarge the scope of the programs with a larger and faster computer. It is also better implemented with fortran 77, in which case the two assembly subroutines can be replaced with fortran 77 subroutines.

The many extra features in the programming demonstrate the applicability in other area.S>f switching theory. The examples selected in appendix II, are typical to its size of on-set number of don't cares, and number of variables. The time used onc-PU is not optimized because of the need of checking out all the steps and understand!nCr the computer withvarious real examples. Further research is directed to develop a whole series of canned programs to do all the logic design on

computers by optimizing the programming.

138

CHAPTER VEXTENSION OF THE MINIMUM MAXIMAL-CUBE THEORY TO HIGHER CUBES AND OTHER KIND OF SWITCHING FUNCTIONS

5.1 EXTENSION TO HIGHER DIMENSIONED CUBES

In chapter II, section 4, we had discussed the distance of cubes higher than 0 . (Ref. to definition 24 and theorem 19). It is natural for one try to generalizing the theory into higher dimensioned cubes and to try the minimization or decomposition of functions having different dimensioned cubes. In chapter II, we studied the interrelationship be­tween cubes and found that not only the distance between cubes played an important part but also the skewness of cubes. The reason we did not use the skewness in setting up distance matrix for 0 -cubes is because the awayness theorems(the­orem 18 and theorem 19) indicated that skewness is always zero for 0-cubes. To handle higher cubes, one has to set up a com­plex matrix C with entries of a pair (d,s) instead of a single number, where d = the distance between cubes, and s = the skewness between cubes. This was followed by a complex matrix D with entries of a pair (f^,fg ) where f^ = the frequency of distances and f = the frequency of skewness. After setting up matrices C and D, one thencan establish the kind of char- ateristics one needs for minimization or decomposition. It appears to be very promising for further research work.

139

5.2 EXTENSION TO OTHER KIND OF SWITCHING FUNCTIONS

Since the whole theory started from a metric space, it is obvious to consider whatever the space and metric of a particular switching functions Would able to render a similar kind of theory like what we have developed in chapter II. In other words, the space Qn can be the cart­esian product of a prime number or the power of a prime number, thus almost all kind of multi-valued logic or Galois switching fuctions are included. However, the metric of the space other than olean is usually not unique and simple, and therefore quite cumbersome'to work with. Recent research indicates that ternarylogics and Boolean Galois switching functions are among the most promising kind other than binary (or Boolean) logic. The metric of these functions has been explored , extensively by coding theoreticians.We speculat from this vantage point that the maximal- cube covering approach will play important role in the coding theory. It is conceivable to discover some new codes or to re-interpret some old codes. In any event, the only limitation of extending the theory of maximal-cube covering is ones own imagination.

5.3 DISCUSSION

It is interesting to note that although the original

140

purpose of developing the algebraic structure of maximal- cube coveringwas to manipulate Boolean fuctions, however it approaches its own mathematical form and structure that warrants the study of its own. Further study in the interest of switching theory or coding theory or just abstract algebra are being pursued by the author of this dissertation; the result reported in this dissertation representing the original goal of the authors research subject.

141

CHAPTER V i

SUMMARY, CONCLUSIONS AND RECOMMENDATIONS

6 .1 SummaryIn this dissertation, we have systematically developed

the Minimum Maximal-Cube Covering approach to handle switching circuits and its extended application into other areas. The work can be summarized as follows:(a) We have looked into the intrinsic characteristics of

switching circuits by chronologically surveying previous work and comparing and analyzing various approaches thoroughly. The results have led to a de­fined set of most desirable characteristics for simplifying and realizing switching functions with the most economical circuits.

(b) We have developed the algebraic structure to facilitate the Minimum Maximal-Cube Covering approach to switch­ing theory.

(c) The application of a branching graphvto two-level minimization of switching circuits was fully illustrated with detailed examples. Other switching problems were also treated with the branching graph method.

(d) An alternative method of using algebraic-geometric properties of the Minimum Maximum-Cube Covering was also developed. The application of this method into

142

switching circuits ^as illustrated with various examples. Since this method turns out to be better to implement with a computer program, demonstrative computer programming was carried out in detail to verify the validity of the intrinsic merits of com­puterization. Because of limited resources, we were able to use only the Univac Spectra 70 VMOS at New Jersey Institute of Technology. No attempt was made to optimize programming.

(e) Exploratory work has been carried into other fields of interest in Chapter Five.

6 .2 Recommendations(a) Due to lack of resources, we were not able to fully

develop the complete set of computer programs for digital system usage in industry and for educational purposes in universities. We would urgently recommend this development using the theory and technique develop­ed in this dissertation.

(b) The extension of the Minimum Maximal-Cube Covering into other fields of interest was only lightly explored. Further work should also be attempted.

6 .3 Conclusions(a) The mathematical properties of the Minimum Maximal-

Cube Covering, in addition to their original purpose of facilitating handling of switching circuits and

143

digital systems, are quite interesting and rich in their own right. In the area of Boolean algebra and the area of lattice theory, Minimum Maximal-Cube Covering seems to offer a very fresh, powerful new tool.

(b) With the ever-increasing availability of microprocessors and LSI's, hardware realized instruments and machines using Minimum Maximal-Covering principle are quite conceivable.

144

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29. Klir, G. J. and I. K. Seidl. "Synthesis of SwitchingCircuits." London: Illiffe, 1968. New York:Gordon and Breach, 1969.

30. Kohavi, Z. "Switching and Finite Automata Theory."New York: McGraw-Hill, 197 0.

31. Kriege, M. "Basic Switching Circuits Theory." NewYork: McMillan, 1967.

32. Ledley, R. S. "Digital Computer and Control Engineering."New York: McGraw-Hill, 1960.

33. Lewin, D. "Logical Design of Switching Circuits."London: Nelson, 1968.

34. McCluskey, E. J. "Introduction to the Theory of Switch­ing Circuits." New York: McGraw-Hill, 1956.

35. McCluskey, E. J. and T. C. Bartee. "A Survey ofSwitching Circuit Theory." New York: McGraw-Hill, 1962.

36. Marcus, M. P. "Switching Circuits for Engineers."Englewood Cliffs, N.J.: Prentice-Hall, 1962.Second Edition, 1967.

37. Mendelson, E. "Boolean Algebra and Switching Circuits."New York: McGraw-Hill, 1970.

38. Meo, A. R. "On the Synthesis of Many-Variable SwitchingFunctions." In Network and Switching Theory, Ed. G. Biorch. New York: Academic Press, 1968, pp. 470-482.

39. Miller, R. E. "Switching Theory." 2 vols. New York:John Wiley, 1965. Volume I: Combinational Cir­cuits, Volume II: Sequential Circuits and Machines.

40. Moisil, G. C. "The Algebraic Theory of Switching Cir­cuits." New York: Pergamon, 1967.

41. Mukhopadyay, A. "Recent Developments in SwitchingTheory." New York: Academic Press, 1971.

162

42. Muroga, Saburo. "Threshold Logic and Its Applications."New York: Interscience, 1971.

43. Nagle, N. T., B. D. Carroll and J. D. Irwin. "AnIntroduction to Computer Logic." Englewood Cliffs, N.J.: Prentice-Hall, 1975.

44. Oberman, R. M. M. "Disciplines in Combinational andSequential Circuits Design." New York: McGraw-Hill, 1971.

45. Phister, M. "Logical Design of Digital Computers."New York: John Wiley, 1958.

46. Prather, R. E. "Introduction to Switching Theory:A Mathematical Approach." Boston: Allyn & Bacon,1967.

47. Roginskii, V. N. "The Synthesis of Relay SwitchingCircuits." New York: Van Nostrand-Reinhold, 1963.

48. Sellers, F. F., M. Y. Hsiao and L. W. Bearson. "ErrorDetecting Logic for Digital Computers." New York: McGraw-Hill, 1968. Ch. 2, pp. 17-27.

49. Seshu, S. "Graph Theory and Network Analysis." Read­ing, Mass: Addison-Wesley Publishing Co., 1960.

50. Sheng, C. L. "Threshold Logic." New York: AcademicPress, 1969.

51. Sheng, C. L. "Introduction to Switching Logic."Educational Publisher, 1972.

52. Simmons, George F. "Introduction to Topology andModern Analysis." New York: McGraw-Hill BookCo., 1963.

53. Soucek, B. "Minimization/Minicomputers in Data Pro­cessing and Simulation." New York: Wiley-Inter-science, 1972.

54. Staff of Harvard Computation Laboratory. "Synthesisof Electronic Computing and Control Circuits." Cambridge, Mass.: Harvard University Press, 1957.

55. Torng, H. C. "Logical Design of Switching Systems."Reading, Mass.: Addison-Wesley, 1964.

163

56. Tou, J. T. "Applied Automata Theory." New York:Academic Press, 1968.

57. Unger, S. H. "Asychronous Sequential Switching Cir­cuits." New York: John Wiley, 1969.

58. Warfield, J. J. "Principles of Logic Design." Boston,Mass.: Ginn & Co., 1963.

59. Wickes, W. E. "Logic Design with Integrated Circuits."New York: John Wiley, 1968.

60. Wood, D. E. Jr. "Switching Theory." New York: McGraw-Hill Book Company, 1968.

Al

APPENDIX A COMPUTER PROGRAMS

1. PROGRAM LARGE2. SUBROUTINE CUBESD3. PROGRAM DISTBL4. PROGRAM DIGBIT5. ASSEMBLY SUBROUTINE NDIST6 . ASSEMBLY SUBROUTINE BICHA

F O R T R A N iv ( V E R 45 ) S O U R C E l Is T I N G :_ 1 _ _ . PROGRAM... LARGE .......... t2 COMMON Cm AR(520,4),m T(520),T1(520),T2(520),OUTA{8)3 % iCUBEN.DIAN,MTH,ZNTH

_ 4_______PJM E.N S I CN._B„(9)j B I. N.OJ 9,9),TEM IJ .104 (J)______________ _5 EQUIVALENCE (TEMT<i >;TJ<i >>6 DEFINE FILE 28<520'52n*E ,RW)7 IMPLICIT INTEGER (A-Z) ...8 1 ”READ(5*2»ENDalOOO)NViDONT»ZErO9 2 pORMAT(313)

.10..... . IF(NV,GE. 9001GO,_.TD 1000 ______________-I NV4 = NV/4•0*0 * 812 CALL NVArjs(NV)13 CALL BICHAINV) ......... - ...........................—14 DO 502 1=1,104015 502 TEMT(I)=0i.6... DO 4_U = i #65...... — ...-. - -----17 VI=(U-1>*1618 VA=VI+119 - .yB=yI+16 - - -- ■ -20 READ(5,3)(TEMT(V),V=Va 7VB>21 3 FORMAT(8I10)PP DO ...4__V = VA1 VB _ _____________23 IF(TEMT(V).LT.2147483640)GO TO 424 Nt EM*V25 . G6..TO 5 .... ..... .................... ........26 ,4 CONTINUE 927 * 5 K = i2 8_______ DO 9 J = 1,NTEM _ ___ _____ _____ ___________________29 MT(K)sTEMT(J)30 IF(ZERO,EQ.0 > GO TO 631 IFCMT(K) .EQ,0)GO TO 9 __________ ____32 6 j6=K-i33 lF<K.Eo.i>G0 TO 834 __ DO 7 I .= 1, J 6 __ _..... . ...-------- --------------35 IF £ MT(I).EO.MT(K))G0 TO 936 7 CONTINUE37 8 K = K +1 _________ _ ..._____3 8..... 9 CONTINUE39 10 NT=K-24 0_______ CT=NL"Do NT______________________________________________41 CUb ENb O42 PRINT ll,NT,CT*DONT,NV43 11 FORMAT ( ' 1TOATAL NUMBER OF ON-SET AND DONT^CARE 5 ._...=44 %, 15,/,* ON-SgT * * , I5,5X, t DONT"CARe S = *;i545 %»/;• numbers of variables = M 5 ).46______ IF(CT.LE,Q)G0 TO 1_ _ _ _ _J____ ____________47 RW-148 MAXCUB=149 DO 12 1=1,10 __________50 MAXCUB=MAXCUB«2

A3

F O R T R A N jv (V ER 45 ) S O U R C E L I S T I N G : L A R G E P R O G R A M51 _I F.tM A X CUB.. G T_i N TJ G 0' 10-1.4.a

12 CONTINUE14 MAXCUB=I"1

54 B I NO(1.1)-1......& “55

Lr A ' V \ w f tmf w —BI NO(2.1) =2

56 BIN0(2,2)=1------ 57---- .. DO i42 I=2»MAXCUB

58 11=1-159 BIN0CI#1>=I

______ 60 • DO-141 J = 2, 11 ----- --61 JlsJ-162 141 BINO(I»J>s BINO(11« J)6 3 <4gBIN0(IiI.)=i64 WRiTc(2,145)MAXCUB, ( ( BI NO ( X, J > . *1 = 1, KAXCUB)65 145 FORMaT(’IMaxIMUM CUgE DIMENSION = ’#13*//,66 _______ PRINT .3 0 */./.! # M.T / I u , 1=1, NT)67 30 F0RMAT(/,2(5X, 'MINT.#' . 14, * - »,U0>>68 Do 45 1=1,NT6 9_____ __CHAR(I»1)=Q _70 CHAR<r,2>s071 CHAR(i;3)=0.72... .- CHaRU-.4) = 0___________ ____ .....73 45 CONTINUE74 ZENsO75 DO 100 12 = 1* NT ..... ....76 DO 51 I=i * 977 B(i)?0 ?

______ 78.._ DD . 6-0 -I-lfi 1 * NT----------- —79 I F ( U,NE.l2)G0 To 5680 MjD = 081 GO TO 6062 56 CACL NDIST(MTD»MT(ID,MT(12))

! 83 lFfMTD.LE.MAXCUB)B(MTD)=B(MTD)*il 84.. IFiMID . LE.9)g O TO . 60______________ _

85 M TD = 986 60 T1< 11) =MTD

......87 WRiTE(28*RW,6_l> (Tit I) . I=1,NT)1 88 61 F 0 R M A T ( 2 0 0117 2 0 6 11,12 0 11)! 89 IFCB(1),Eq .O)GO TO 85! on R1=MAXrUR ____ ____

91 B2=B1*192 DO 80 Jll=l*Bl93 J1=B2-J11r 94 DO 70 J2=l*J1

i 95 IF(B<J2).LT.BINOCJl»J2)>GO TO 8006 7n CONTINUE .. _ ____________97 CH AR (I 2 * i )=J198 IF(B(1).NE,BIN0(Jl,l>)GO TO 7299 CHa R(I2*2)=1

100I GO TO 100. .. .... ....

;i=17maxcub)(14 I § ) )

A4

F O R T R A N IV (VER 45 ) S O U R C E L I S T I N G : L A R G E P R O G R A M 72 CHAR (12,2)=5 „ ... .......

102 Go TO 100103 80 CONTINUE10 4____ 85 M.INT=.MT( 12)105 PRINT 90,12,MINT106 90 FORMAT!'OMINTERM #*,I475X , ' - ' ,110,’ IS ISOLATED AND ESSEN107 ioo continue .........108 PRINT 98109 98 FORMAT{'1THE CUBE COvERING LIST :*»//>

_ 110 - - WRITEII2# 101 > I ( cHAR (11 J ), j sl » 2), I =1 , N 1) ___________ ________ _111 m i FORMATCnTHE CHARACTERISTICS : ' , / , ( 5X , 2 2 I 3 > )I I I CHECK=0113 IF(CT,GE,NT)GO TO 102 _______ _______ _____ _ ____114 DO 103 I =1» DONT115 X=CT+I116 .... CHAR(X» 31-9.999_____________ - - - __ _________117 CHAR(X,2 >=109118 103 continue119 102 M s 2 * * M a XCU R.. _ _ _.-...... - -....... ■ — ---- -------120 KOUNT=0121 DO 105 I_1,NT122 IF(CHAR(i,D .NE,MAXCUB)GC TO 105 . ........ __123 k o u n t =k o u n t +1124 105 rONTjNUE125 If (KOi|NT.GE,MAX)GO TO 107......................... ....... ....126 DO 106 1=1,NT127 , IF(CHAR( 1,1) ,NE,MAXCUB)GO TO 106

1 C.HAR( I ,1)=CHAR ( I ,4 ^ 1 __ ________ _______________ _____________129 CHAR(1,2 ) =5130 106 CONTINUE131 MAXCUB = MAXCU8-1 _ _ _________ .......132 WRITE(2,io8)MAXCUB .133 108 FORMAT!'oMAXIMUM CUBE DIMENSION HAS BEEN LOWERED TOJSIlO134 .. Go TO 102._ .____ ________________________________ ____135 107 CONTINUE136 IF(ZEN.EQ.l)GO TO 125137 DO 120 16 = 1,NT ____________ _______138 IF!CHAR(16,2),NE.i)GO TO 120139 D=CHAR(l6,l>14 0________MTH= I 6_____________ _____ ___ ____________________________________141 ZNTH=1142 CALL CUBESD(ZNTH,MTH,D)143 IF!DIAN,NE,0)GO TO u 9 . ___________ ____144 CHAR ( MtH »j.) SD**1145146

CHAr (MTh *2>=5If .(P J.NE.. MAXCg B)G 0_ j 0_12 0

147 ' IF!CHECK.NE,0)GO TO 120148 GO TO lo2149 119 IF(D,NE,MAXCUB)G0 TO 120150 IF! CHECK.NE .0)GO f0 120

A5

F O R T R A N IV ( VER 45 ) S O U R C E L I S T I N G S L A R G E P R O G R A M

-151-152153 J.5-4-.55

CWEC-K-rI —.120 CONTINUE

ZEN = 1 WR-i-T E (-2.12 21MAX CU B.122125

-14.0

15P 156 157.15B 159

_160~161162

-1631641651 6 6_______167168169 . ___170171172 — —

173174 160175176177 150

. 1 7 8 _______179180181 1701821831 8 4 ------185186187 .....188 189

-19Q .__.l7-5. l’i i’ 2

.. 193____180194195

. 1 9 6 ________197 2 8 0198199 3002 0 0

FORMAT('nALL CORE CyBES HAVE BEEN SEARCHED.' AND THE MAXIMUM CUBE DIMENSION IS : *715)

g l s M A x O O g .......................B2=B1*1DO 280 Jllsl1BlJ1 = B2-J11_____________ __ ... .......... ..... ...DO 180 J2=1»NTIF(CHAR(J2.1I,NE.Jl)GO TO 180 IF(CHAR(J2 »2>*NE ’5 )gC TO ^ 0 ...PSuApE = CuAR (j2» 3)CLOSE=CHAR<J2.4)MTHsJ2 __ .....D- JlDO 170 16=1.NT. IF j[ J 6 »EQi J2 ) GO TO 170 ..................... ....IF<CHAR(I6*l>.NE.J!>GO TO ^ 0 IF(CHAR(l6*2>.NE*5>G0 TO <7 Q

IF (CHARI I & .3). GT ..P 5 H A R E ) G 0 10 4 7 0 ........ ...IF(Ch Ar (16,3),EQ.PSh ArE)g O TO 150 PSHARE=CHAR(16.3)M T H s I 6..GO TO 170 CONTINUEIF(CHAR(16.4),GT.CLOSE)GC TO 170...............CLOSE=CHAR(16.4)Go TO 160CONTINUE__________ _ ................ ............ZNTH=5CALL CUBESD(ZNTH,MTH.D>I F.( D I AH. NE« 01GO..T0 175_________ -_________CHAR(MTH.i>=D-l IF(Ch ECK.NE,0)g O TO 180If (D,NE.MAXCUB)GO TO 180 .CHECK=CHECK^1 GO TO 102

_I.FiD.NE.MAXCUB)G0. TO .180_________________ ______IF(CHECK.NE.0)GO TO i8o CHECK=ic o n t i n u e . .. ______DO 280 J6=l1NtIF(CHAR(J6.1),NE;jl)GO TO 280.IF(CHAR{J6.21 ,EQi5)GO. TO .14 0_______ ___________CONTINUEPRINT 300,(CMAR(I,2)iI=liNT)Fo r m a t c o t He f i n a l c h e c k i n g ',/,.(iOX,l2l5j )GO TO 1

A6

FORtRan iv (VER 45 ) SOURCE L IS tING S LARqE PROGRAM 201 .1000 STOP .. __.. _ .......202 END

r iLr

A7

FORTRAN ly (VER 45 ) SOURCE LISTING* CUBESD SUBROUTINE 12/16,77 2o:?0..SUB H OU Ti N E. -C U B ESBT ZNT.H*MIH.»D ).

9 IMPLICIT INTEGER <A,Z) . . . . i3 COMMON CHAR<520,4> »MT(520> »Tl (520> »T2<520 ) #OUTA(B)TNV4;nt;rm'

A — % > CUBEN , dI AN , mTh* ZNlu--------------------------------------------- 15 DIMENSION T(520),s(520)6 DEFINE FILE 28<520,520^E ,RW>7 ... __ CHK"2*#D ... . __8 RWsMTH9 REa D(28*RW.125) (TlU )» I=l7NT )

125--FORMATt200.11;20011*12011, - — .- - ---------- -- ...11 MINT-MTCMTH)

IF(dTNE,9)G0 TO ,4013 DO ,30 I^.NT14 IF(TKI) LT 9)G0 To 13015 Ca l L NDI§t (Nt D»MINt »Mt (I))

-16___ __ T l j l ^ M T D ................................. -........17 130 CONTINUE18 140 IR_MTH1 9 _____ D I A N = Q ....... - -________________________ — __ _____ .....20 RAWMT=021 S3=NT+1.22....... CHAR (S3* 4 ) =9000___ . _.-._------ -.....~23 T ( i ) = I R24 00 200 J3=1'NT25 ...... IFJTKJ3) ,NE,D)Go Tq 200 . .26 IF<CHAR<J3»l).LTvDJGO TO 20027 I RW = J3 f28 ... READ {28 ’.RW *125 )Tl2 (I) * I=1* NT) .... ..._...29 T(2)=J330 IT=231... ... Do 160..J4 = 1,NT ...... . - - .-32 IFfTlCj4),GE,D )GO TO 16033 IF(T2(J4),GE,D )GO TO 1603.4________LDsTl (J4) *12 (J4)______ _...______ _______ _35 IF(LD.NE.D>GQ TO36 ITsIT+i37 ......T £ i T ) = J4...................... . . ..38 160 CONTINUE39 1F (CHK,NE,IT >GO TO 200

. 40 .______ WRI TE C 2j 162)MT (.IR) #.ML( J3 )_________________________________41 162 FORMAT('oCUBE C'.IlO*’ »'»IlO»l > H*5 BEEN FOUND.’)42 ISs IT4 3 ______ DIAN = 1._____________ - . _____ __ _- - ■ ■ - ■ -?44 IF’(ZNTH,EQ,5)G0 TO 16545 IF(ZNTH.EQ,1)S3=J3.46_______GO T0..300_____________________________________ -.. .47 165 COUNT = 048 Do 170 1=1, IS4 9____ TE = I U 1 . _ ........................... - -50 Ip(CHAR(TEi2),GTil0)GO TO 170

A8

FORTRAN IV (VER 45 ) SOURCE LI ST!NG J CUBESD SUBROUTINE 51____ COUNTs.COUN.T + 1 ........... .. ... ............. . ........

52 i7o continue53 IF(COUNT.l T.Ra w MT)GO TO 2005 4_____________D U.N T_ i E 0 i_R A W ML) G 0. ..T D 175________________ ____________55 172 S3-J356 RAWMT=COUNT57 DO 180 I =1, I S . .. .58 S (I> =T( I >59 1 BO CONTINUE

.... 60_______ GO TO.200_________ ... -______ - .— - — .......61 175 CONTINUE62 WRITE(2,173)MT<S3),MT<J3>

_ _ 63 i7J FORMAT (’q WATCH FOR '»Ii q #' AND * * 11 o) . _____ __ 64 IF(CHAR(J3,4),GT,CHAR(S3;4j)Go TO 200

65 GO tO 172 66 .200 CONTINUE ._ __ — .

67 IF(DIAN.EQ,0)GO TO 70068 205 RW=S369..........Re AB<.?SIRW, 125 >.< T2 U I * 1;NT > ________ _ _________70 DO 210 1=1# IS71 TFbS(I%

. 7 2 _______CHAR1J.F.#2)>CHAR(TF#2)*10 0 . „ .........------ ------73 CHAR(TF » 3)=999974 210 Continue7 5 _______DO...220 I =1 rN'T .... . . .. ... -......... - - -... -76 IF(CHAR(I.3L),EQ.9999 )GQ TO 22077 f WABsTi(I)+Ti2(I>-D

_ 7 8 _______ CHAR.Ui4>=WAB + CHAR< I,4>...... -....... -79 DA=CHAr (i »1)80 If (DA,l T.WAB)GO t O 22081 GsDA~WAB82 GlsG83 IF(G.GE,D)G1=D

84_______ SHARE = 2«*G1 . . _.. -.... -----------85 CHAR(i;3)=CHAR(I,3)+SHARE86 220 CONTINUE87 WRITE(2#212)(CHAR(I,3)71=1.NTj _____ ______88 212 FORMAT(tOCHECKING POINJS»,/,<10X,12 I 5 ) )89 WRITE<2#2l6HCHAR( 1,4). I = i»NT)

_ 9 0 21 6_E0r m A112JD. AC CUM.U U A LE D....AMABj e S S_LL,‘ZZ^ < 10 X.,1.215 ) )--------91 CuBEN=CuBEN*l92 PRINT 228. CUBEN .....93 228 FORMAT(///#1OTHE »,I5,i-TH CUBE COVERING IS :')______

. c A L L BIC H A 2 < M t ( IR j, M T ( S 3 ) 7 0 U T A >95 PRINT Z30.d .MT U R > » MT(S3), t OUTA < 0 ), 0 = 1 ,'NV4 > „

96___ 230_F0_RMATj5X,J4,_»■-CUBE...*_ i ' 7 110 # _ » ’ /J10..J L=.i» 3X.*8A4 j97 WRITE(2»222)MT(I R)»MT(S3)98 222 F0RMAT('oCUBE < *• 110 * * •*'110'f > IS SELECTED.')

SSL_Do..232 1=1, IS _ ________„ ...100 STE'msS CI >

F O r T r A N IV (VEr 45 > S O U r C E LI ST I n g : C U b E S d s u b r o u t i n e-101........S< nsKTCSTEM.)________________________ _________________— -102 232 CONTINUE103 PRINT 2 3 5 , < S U > p I = 1,I5>

__144___ 235—F-oRMA T-( -0 T H J 5 £ UBg—Co V g R S THeSE -M-I -NT-eRMS-7-L,/^-(5X£6|l0 34105 IF(ZNTH.EQ,5>G0 TO 700106 1 F (ZNTH.NE «1)GO TO 68010 7 PRINT 240, IR.M.KJR> .. . - v — -108 24o FORMAT ( ’ qTHIS MINTERM MT( ’ » I 4 * ’ ) = *»Iig»5X*MS A CORE109 G0 To 700

.110___ 300 D O -320 1=1 * IS - ------ — -------------111 S (I)=T(I)112 320 CONTINUE113 GO TO 205114 680 PRINT 685115 685 FORMATt’OTHERE IS AN ERROR IN THE PROGRAM!!!’)

...1.16 7ob return ....... — -------------------- ----..... — ... ....117 end

A10 .

FORTRAN IV (VER 45 > SOURCE LISTING*1 PROGRAM DISTBl2 COm m ON m T(520 ) ,T1(520)IT2<520 > »OUTA(B) ,NV4«NT ,R_W3 x .c u b e n .d i a n .m t h .z n t h4 DIMENSION B (9)iTEMT(1040 )

E 0 UIV AJ_ ENCE-1T IMi.i I jI a6 IMPLICIT INTEGER fA«Zl7 1 READ (5,2,END=100D>NV,D°NT,ZER08 2 pORMAT(3 13)9 IF(NV,GE,900)GO TO 1D00

10 NV4»NV/4.0*0»8.11 . _ _ CALL NVArjs<NV)- - -12 CALL BICHA(NV >13 DO 502 1=1.104014 502 TEMT(I)=015 DO 4 U=1.6516 VIp (U»1)*1617 vAsvItl.- ....18 VB=VI*1619 READ(5,3)(TEMT(V),V = VA,VB>20 3 Format<i6 i5> . _____21 DO 4 V=Va .Vo22 IF(TEMT(V),LT.90000)GO TO 423 ...... NTEM = V ...... .......... ..........24 Go TO 525 4 CONTINUE26 5 K = 127 DO 9 jal.NTEM28 MT(K)=TEMT'( J)2 9 IF (ZERO, EG). Q> GO TO 6...........30 IF(m T(K>.EQ.OJg O to 931 6 J6=K~132 IF(K.EQ,l)GO TO 833 DO 7 1=1.J634 IF <MT CI),EQ,MT(K))Gq TO 93 5 7 CONTINUE__________________________36 8 K=K+137 9 CONTINUE38 10 NT = K-239 CTsNT-DONT40 CUBEN=04 1 PRINT.il,NT,CI.iPMT,N.y42 ’ ”ll 'FORMAT ( 'lTOATAi NUMBER OF ON-SET AND DONT»CARES = »43 %,15,/,’ ON-SET = »,I5;5X. • DONT-CARES = *71544 NUMBERS OF VARIABLES = M 5 )45 IF(Cf,LE,0 > G0 TO 146 DO 100 12=1,NT47 PRINT 51,12, MT( 12) , - -48' 5i FORMATC'oTHE DISTANCE FUNCTION FOR MINTERM49 %.1 IS *',//>50 DO 52 1=1,9 _____

All

FORTRAN IV (VER 45 ) SOURCE uIsTING: pISTBL PROGRAM 51 52 B < 1 ) =0

52 ___ DO 60 11 = 1,NT__ ___ ______________ ___ ____________53 IF(11, N E ,12 > GO TO 5654 MTD=05 5_______ r Q TO 60_________________________ _______________________________56 56 CALL NDIST (MyDiNT(11),M j M 2> )57 B (MTD)sB(MTD)*158 IF(MTD,LE.9}G0 TO 6059 MTD=960 60 Tl(I1)=MTD

61 PRINT 61, ( ( I |J1( I ) ), I s 12 • NT >____________ . -..62 61 FORMAT<5(2X, »T< ' , M , ’ ) = 'ID,/)63 PRINT 62,(B(I>iI,l»9)64 62 FORMAT(//,' THE DISTANCE DISTRIBUTION ARE J ',/,9 I 5»/////)65 ioo c o n t i n u e66 go TO 1

67 1000 STOP ........ . .......... ........................ .... ...68 END

ii

A12

F O R T R A N IV ( VER 45 ) S O U R C E L I S T I N G :1 PROGRAM DIGBIT2 COm m ON Cu a R(520,4)#m T(520),T1(520>,T2(520),OUTa <8);NV4.Nt d..3 % , CUBEN VD I AN~, M f ff, Z N T H * -H.4 DIMENSION B(9)#BINO(9,9),TEMT(1040)5 EQUIVALENCE ( TEMT (1 > » Tj. (1 > >6 IMPLICIT INTEGER (A - Z 5 ' ;7 1 r EAd (5,2,ENd =1000)Nv ,d ONT,ZEr08 2 pORMAT(313)

9 IF(NV.GE,900)GO TO 100010 NV4=NV/4.0+0;8

11 CALL NVAr IS(NV> _............ ..... ....12 CALL BICHA(NV )13 DO 502 1=1,1040

14 502 TEMT(I)= 015 DO 4 U = i»6516 V j = (U-l)*16

__ 17 _YA = VIt_l_ __ ....... ..... ..............18 vB=vI+l619 READ(5,3) (TEMT(V),V = Va VVB)on 3 FORM A T (j6 I 5 )21 ' DO" 4 V = V A » VR22 IF(TEMT(V) LT 90000 ) Gg T0 423 __Nj EM = V ' _ __ __ _________24 'GO TO 525 4 CONTINUE26 5 K-l27 . DO 9 J = 1» NTEM ...28 I MT(K)s TEMT({J)

29 __ IF(ZERO,EO.O)GO TO 6 ______ _____________________________ ___30 T fT mtTK") . EO . 0 ) GO "TO 931 6 J6=K-132 IF(K.EQ.l)GO TO 833 DO 7 I = 1» J6 ..................... ........ ......34 I F ( MT ( I ) .EQ'.MT(K) }GO TO 935 7 CONTINUE________________________________________________ ___ ____36 8 K = K + 137 9 CONTINUE38 10 NT = K-2_____________________________________________ ____

39....... CT = NT-DONT ...... ....40 CUBENsO41 PRINT 11, NT > CT»DONT»NV _ ______ __________

42----H~F'0R M'A T r'TT’OXT AlT N U~M B E R~~DF~‘DN^ SET“TN D~~D~0~NT# CARES = '43 % , 15,/, ' ON-SET = *,15 # 5X, » DONT-CARES = ’.1544 numbers of variables = *,I5> _ _ ...

"45 I F ( CT . L~E , 0 ) G 0 "T 0 1 “46 DO 21 1=1,NT47 CALL B I C H A K M T U ) ,OUTA) ....................4 8 “ PRINT 2 0 * I » M T ( IT, ( 0 U T A ( JT7TT= ! 7 W 4 )49 20 F0RMAT(5X,’MIn TERM #',14,' = ',14,' IS »,5X,8A4)50_ 21 JION't I NjjE___________________________________ _ ......

Or Tr AN jV (VEr 45 ) SOUr q E L jST jNg : DIGb It Pr Og r AM51 GO TO 15 2 10 0D_ STOP53 END

A14

*«

1 .

TITLE 'N d IS T-- A b INa RV DISTANCE FINDING FUNCTION— -OCT. FOR FORTRAN PROGRAM"! FOLLOW THE FOLLOWING TWn STEPS:

A L y i n g t h e # of v a R iabi'e s ,n , by c a l l n v a r i s c n )

7 5 '

* 2 «

(NOTES THIS N WILL BE USED UNTIL NEx T CALL N VAR IS. DEFAULTTlF^ FINDING THg DlSTAwCg QF 2 VICTORS,I,J, BY FUNCTION NDTST(IfJ)

NVARIS

NDIST

SPACECSECTUSINGLLST . LNR ASTH M V j BRSPACE.ENTRYUSINGSTHlmLLLXR

NVARIS,15 1 *0(0*1)1» 0(0*1*1 *NV1.1 1.N321.PRESHF+? 12(13),> 'FF* 143 _ _______NDIST N D I S T * 15 1,5*_24(1_3 i3 ,4 ,aT±)5 » 0 ( 0 »11 -2»_o(~0-*3 )_____3 * 0 ( 0 ' 4 5 3 ■PRESHF 1 SLL 3/D

SLR lilL 4 * NV

LP SLR 2.2SLDL 2,1AR 1,2BCT 4‘,LP

1 ST 1, 0 ( 0 , 5 )| LM 1,5.24(13)

M V I 12(13) 'FF'BR 14SPACE 3

N32 DC F ' 32'NV D C " F ' 3 2"' ...........

end/EDN

A15

B ICHA

BICHA1

SETNiP.RESHFlLP1

BICHA2

SETN2PRESHF2PRESHF3LP2

TITLECSECTUSINGL ____LSTSTC

’BICHA — b ITq TO CHARACTERS (FOR OUTPUT)B I C H A * 1:5 1.0(0*1)1 *0 (0 »1 )1»NV1. SEt N:1-*1

STC1_NRASTHSTHSTHMVIBRSPACEENTRyUSINGSTMLLLM VC LSLLSLRSLDLAHs f c..ABCTMVILMBRSPACEENTRYUSINGSTMLMLLLL ____MVCXRSLLSLL 'SLRSLDLSLRSLDLALRBCAH

1*SETN2«11,11,N321,PREoHF1+2 1.PRESHF2+21 . PRESHF3 + 2 12(13);>'F F ' 143BICHAl BICHAl.15 1 * 1? 2.4 (1 3 )3 » o ( 0 • 1 }3» 0(0*3)1 f ..4.io' 1 > . ..0 ( 32* 1 1 * Bi ANK «,NV3.0 ___2,2 2.12,ZERO %, 0 (0 ,1 ) l.ONE4.LP112(13),>'FF* 1,4,24(13)14_3 ....BICHA2 BICHa 2 ,15 1,6,24(13)3,4 * 0(1)3.0(0*3)5,0(0*4) 1 .8 (0 *1 )6.NV0(32,1) ,Bl ANK5,33.05.0 -----------2 . 22.1~47T4,14,4"57DTCF 2,ZERO

A16

STC 2,0(0*1)B ADDRDASH MVI 0(1), Cr- *

ADDR A 1 .o n eBCT 6.LP2LM 1,6 * 2^(13 >

----------- MVi__ 1.2 Li z ) . ; > . ± F F lBR 14

ZERO D1" X ? 00F0 '__ONE DC F '1 *

NV DC F 1 32 *N32 DC F ' 32 *

_b l a n k Dcend

. __CL 3.2_!__•... . .

r

t ............. - ...... - -------- ' -------------------1

APPENDIX B

EXAMPLE OF 4 VARIABLES

A18

t oatal n u m b e r of o n -set and d o n t#c a r e s =ON-SET = 8 DONT-CARES = 0NUMBERS OF VARIABLES_ =_ ___ 4_

MINT,# 1 = 0 MIm j,# 2 =MINT.# A - 7 MINT.# 5 =MINT.# 7 ' '=.ic M l Nj", #'.......S'"" '="

THE CUBE COVERING LIST :

THE 1-TH CUBE COVERING IS :_ ... ..1-CUBE s (____ 8 , 10 ) =______1 U- 0

THIS CUBE COVERS THESE MINTERtfS {8 10

THE 2 -TH CUBE COVERING IS s. 1 - CU B E . = _ (___ .14...,... ...15 __) j=___ 111-

THIS CUBe COVERS THESE MI\TERKS 1_ 14__1 5 _________ _ ____

THE . 3 -TH CUBE COVERING IS 51-CUBe = < 7 % 5_ .. L =______ ° 1 ' —

THIS CUBE COVERS THESE MIN TERMS :7 5__ _ ... ____

THE 4 -TH CUBE COVERING IS :1-CUfE = ( 1 #___ 0 I s 000-

THIS CUBE COVERS THESE MINTERMS !1 C

THE FINAL CHECKING105 105 105 105 105 105 105 105

ToATAL NUMBER 0F ON"SET AN° CcNT*CARe S = 9ON-SET = 9 DCNT-CAR5S = 0NUMBERS OF VARIABLES = ____ 4_____________________________ ..

MINT,# 1 = 0 MINT.# 2 = 1 MINT.# 3 = MINL-JL 4 = 5 MINT.# 5__=______7 MINT..# .. 6 . 5 ..

MINT,# 7 S 10 MINT.# 8 = 14 MINT.# 9 s 1

8

1 MINT.# 3 s 515 MINT.# A = 148' MINT,#- "

U7 CD J*

A19

THE CUBE COVER I r.'G LIST :

THE 1-Th CUBE COVERING IS :2-CUSE = ( i , 4 ) = o-0-

THIS cube covers these kinterms 1 ____ 1 4 0....5THIS MINTED MT( 2 ) s 1 IS A. CORE POINT

THE 2 -th C■ BE COVERING is’ :1 - C 0 C E = ( 10. P ) - 10-0

j THIS CUBe CcVgRS THESE mInTeRM.S :! 1 0 8

THE 3-Th CUBE COVERING. 15 :I 1-CUbE = ( ?,4 r - 15 ) = 111-

1

THIS CUBE CODERS THESE MIN'TER^S * 14 15

THE 4 - T ►« CUBE COVERING I S :.....1-C^rE s ( 7 , 5 ) = 01-1

THIS CUBE COVERS THESE MJN'TERMS j

7 5THE FINAl^CHECkT n G

105 101 101 205 103 105 105 105 105

A20

TOATAL NlJMOER CF Cu-SET AK‘D DCNT«CARES = 14ON-SET = 14 DONT-CARES - DNUMBERS OF VARIABLES =_____ 4______

MINT , # 1 = 1 MINT .# 2 = 2 MINT.# 3MINT, # 4 = 4 MINT.# 5 - 5 MINT.# 69MINT . # 7. 7 MINT .# 6 -

.. ~ M I NT". VMINT , a 10 = 10 MI\'T .# 11 - 11 MI NT . # 12MINT 13 = 13 MI NT . # 14 - 14 MINT.#

THE CUBE COVERING LIST :

!i

THE 1-TH Cl BE COVER ING IS :------------ . -----------

2 - C U - E = ( 14 . S ) = 1 - - 0

Th IS CUBF C3VEPS Tr)ESE ^INTERw S : 14 6 1 C 1 2

THE 2-TH CUBE COVERING IS :_______ 2-CU3E_=.(_...1 13 ). =.... -“01THIS CUBE COVERS THESE MJMERMS ; _____ 1 ... 13.... 5... 9__________ _.... _

«

THE 3 — T —‘ CUBE COVERING IS :_....... 2-Cv'L 2 = ( 7 ,.....2 ) = G-l-| THIS CUBE COVERS THESE MINTERMS j!........... -7-...2.._._2_____6 ______________________

THE 4 -TR< CUBE COVERING IS :________ 2-cUBE = C 11 .j _£___> _ = _ ... 10--THIS CUBE COVERS THESE MINTERMS ;

H P 9.. 10_____________ ___ ____

K> O OJ

A21

THE 5-Tr- CUBE COVERING IS :2-CUcE z ( 4 ..#___ lf_._) = -1-0

THIS CUBE COVERS THESE MINTERMS j_________4___ 14____ _ 6___1 2_____ ___ ____ _____________ ___________ ______rHE f i n a l CHECKING105 105 105 105 10 5 205 105 205 2 0 5 205 105

105 205

TOATAL NU^PER CF Cn-SET and d c m t «car es = 9ON-SET = 9 DOMT.CARES = pM J M eP? cr v ART A ELrS = .. * .............. .....................

MINT.# 1 = 1 MINT.# 2 = 2 MINT.# 3 = MINT.* =______ 5.___ m I s L. #____ 5_..... 15....M1.NT, 6 ...=..

MIN T . it 7 = 0 MIK T . # 8 = 9 MIN j . # 9 =

THE C U S E C C v E k I N G L I S T :

THE 1 - T H CUBE COVERING IS :1-CuEE.. s < . 2 . 3 ) = 001-THIS CUBE COVtpS t h ese MlNTEppS J .._. . . 2 . .. 3...._.......... ..... .............. ....THIS MINTERM MT( 2 ) s 2 IS A CORE POINT

the 2 -th cube covering is:1-cube = ( 5 , 1 ) = 0-01

THI S C U Bp cn'v ERS "THESE Mi Nt E R MS"~S .5 1

THIS MINTERM MT( 4 ) S 5 IS A CORE pOINT

A22

the 3 - t h c u b e c o v e r i n g .is : .........1 - C U R F = ( 1 4 # 1 5 ) = 1 1 1 -

THIS CUBo COVERS THESE . MI.NTER.MS ».......1 4 1 5

T.HIS M1NTERM. MTi .. 6 . ) = 14.. _ IS A CORE POINT

the *-TH CUBE COVERING IS :1-CUBE - I . . 9 i = 1 0 0 -

THIS CL'Bp COVERS THESE MU'TERMS !....... « 9 .... ............ ..THIS M1NTEEH NT( 7 IS A CORE POINT

THE 5 -th CUBE COVERING IS :2 - c U 3 E = ( 1 1 # 1 ) = - 0 - 1

this "cubf" c o v e r s t h e s e n ' i n t e r k s :1 1 1 3 9

THE FI-T.'AL CHECKING *2 0 5 1 0 1 ' 2 0 5 1 0 1 1 C 5 1 0 1 1 0 1 2 0 5 1 0 5

TOATAL NUi-EER OF CN-SET AND nONT«CARES = 12ON-SET = 1 2 DONT-CARES = 0NUM.BERS OF VARIABLES = 5_________________________ __________

MINT.# 1 = 0 MINT.# 2 = 1 MINT,# 3 = M INT.# 4 -______5 MINT.# 5 = 6____M_.IiVI, #___ 6 . .?

MINT.# 7 - 24 MINT.# 8 s 25 MINT,# 9 5MINT.# 10 = 28 MINT.# 11 = 30 MINT,# 1? r t-j

no

A 2 3

T H E C U B E C O V E R I N G L I S T :

THE 1-TH CUBE COVERING IS :1-CUBE = ( 31 , 30 ) 1 1 1 1 -

THIS c u b e COVERS THESE HIN'TERHS j3 3. ... 30.

I THE p-TH CUBE COVERING IS sL. 1-C'JEE = ( . 28 ., 24 )

THIS CUBE COVERS THESE MJNTEPMS : 2 6 2 4

1 1 - 0 0

THE 3-TH CUBE COVERING IS :1-C'JsE = ( 27 , 25 )

t h i s Cu be c o v e r s t h ^se m i n t e r m s : --- 2.7...2 5 .......... _..........

n o - i

THE 4-th CUBE COVERING IS :1-CUBE. = ( J , 6 .... ) = 0 0 1 1 -

THIS CUBE COVERS THESE MR'TERMS j 7 6

THE r -TH CUBE COVERING IS :________ 1-.CUBE = ( ___5 , 1___) 0 0 - 0 1

THIS CUBE COVERS THESE MIN'TERMS ;_________ 5____ 1-_____________________

A24

THE ft-TH CU3E COVERING IS : ;____1 -C^BE = C .. 0 » . P . > = oop-oTHIS CUBF COVERS THESE MINTERMS :

0 2 .................the FINAi CHECKING_________ ' ___i0 5 .. Ip5_ _ 105.. 105 105 105 105 1D5 105 105 105 1 0

TAL NJU r E R OF CN-SET and Po n t *CARES = 16•SET = 4 DON'T-CARES = 12■ BERS CF Va RJABLFS = ____5 ... . • -. . - - -... ------ — . --MINT.# 1 = 8 MINT.# _ c. = 14 MINT.# 3 - 16M I NT . # 4 = 19 MINT.# 5 = 2 0 MINT.# 6 s ?1MINT . # 7 = 22 MINT-# e = 23 MINT,# 9 - 24MINT • -fi 10 25 MI h’ T - # ii = 26 MINT.# 1 ? = 27MINT .# 13 = 28 M I N'T . # 14 = 29 MINT .# 15 = 70MINT . p 1 8 31 MINT.#

the cube c o v e r i n g list I r

THE 1-TH CUBE COVERING IS :1-CUBE = ( 8 * 24... ) = _ -1000 ...

THIS CUBE COv ERS THESE M I N T E R M S ! _ ' 8.. 24 .......THIS MINTED MT( 1 ) = 8 IS A CORE POINT

THE 2-TH CUBE COVERING IS :1-CUEE = ( 14 , 3 0 ) = -1110t h is cube c o v e r s these mi n t e r m s V .14 30THIS M INTER-' MT ( 2 ) = ' IS A CORE POINT

A25

THE- 3-TH CUBE COVERING IS :2-Cl'BE = ( 16 , 2 8 ) = 1--00

t h i s cube c o v e r s t h e s e m j n t e r m s :16 28 20 24

.THIS MINT.ERM..MT ( 3 ) = 16 IS A CORE POINT

the 4-th cube c o v e r i n g is : 2-CUUE = c 19 f .. . 31 ) = lT-11 .. .

THIS cC3b COVERS Th EFE H jNTc RHS 51? 31 23 27 .... .....

THIS M I N T E V M T ( 4 ) = 19 IS A CORE POINTthe final c h e c k i n g101 101 101 101 2n 9 109 109 209 309 109 109 ?!

2 0 9 1 0 9 2 0 9 20u

TQATAL Nij'-'-ER 0F C' - S E T and dcnt* c ares = 17ON-SET = i': D0NT_CARI­S = 7

________NUMBERS 2_F VAR IABL LS =... 5-----... .....---------------------------

M I N T , # 1 s 0 m i n t .# 2 = 3 MINT.# 3 s 4MINT,# 4 = 6 . Mlf;Tt# 5 = 7 M I N _6 s 15MINT.# 7 = 21 M I V T . # 8 ' = "" 2 3 MINT ,# 9 s 26MINT.# ln = 28 MINT .# 11 _ 2 MINT.# 12 eMINT,# 1 3 = 12 MINT •# 14 14 MINT.# 15 = 24MINT.# 16 = 31 m i n t .# 17 17 MINT.#

THE cube c o v e r i n g LIST :

THE l-TH CUBE COVERING IS :2-c UFe = ( 3 , 6..) = 0O-1- _ . .

THIS CUBE COVERS THESE MINTERMS •________ 3 6 ____ 7____ 2 ______________________ .THIS MINTERH MT( 2 ) = 3 IS A CORE POINT

A26

THE 2 -T.N C'JSE COVERING I S' :1-CURE = ( 2 b , 24 ) = 11C-C

THIS CUBE COVERS THESE MINTERMS :26 24

THIS MINTERV MT( 9 ) = 26 IS A CORE POINT

the 3-th cube c o v e r i n g is :2 -CoBE = ( 28 , A ) = -1 - 0 0

THIS CUBE COVERS THESE m ]NT&h m 5 :2 8 8 1 2 24

.THIS M I N T E !•■ ", M T C 10. ) = 2c- IS A CORE. POINT.

THE 4-T* CUBE COVERING IS :2-CLEE = ( 0 » t ) = G0--C

THIS CUBE COVERS THESE MINTERMS ;6 4 ? ___

I »

THE 5 -TK CUBE COVERING IS :?-CUtE.. s ( 15 . n ) = 0-11-

THIS CUBE COVERS THESE M J N T E P m S :________1.5. . _.6____ 7___ 14 .. _ .

THE 6 -Tr CUBE COVERING IS :_________2-cURE (_____2_3.._»_15__> r _ .--HI_________________THIS CUBE COVERS THESE MINTERMS S

.23 15 7 3.1.. ................THE 7-TH CU3E COVERING IS :_____________________l^C LiB i.... = J._______________ 21_*_.17_____LJL______ 10-01 _________ ________________________THIS CUBE COVERS THESE MINTERMS !________ 21___ 17._________________________________________ — .-----

THE FINAL CHECKING1^5 101 105 205 205 205 105 105 101 101 105 105 205 20 9 2 09

l r 5 id

A27

T0AT/ L fvUvi Eft Dr CN‘-SET AND nCM*CARES = 14ON-SET s 14 DONT-CARES = 0NUMBERS nr VARIABLES _=____5 ______________________MINT.# 1 = 0 MINT.# 2 = 2 MINT,# 3 5 3_M I N T , j 4 4 m i n t .# 5 8 m i n t :# r 10MINT » # 7 11 MINT.# 8 12 MINT.# 0 - 16MINT.# 10 20 MINT .# 11 24 MINT.# 12 = 26hint•# 13 .. =____27 .. M 1 fc T . # 14 = 28 MlwT.#

THE CUEE COVERING LIST :

THE 1-TH CUBE COVERING IS :2-CUBE ? ( 3 10 ) = 0-01-

THIS CUd F COVERS t h e s e MINTERMS S 3 10____ 2.. 11___________

THIS MINT£RH MT( 3 ) = 3 IS A CORE POINT

y

THE 2-TR CUBE COVERING IS :3-CUBE = ( 4 * 24 ) = 00

t h i s c u b e ' c o v e r's ' t h e s e h i n t e r m s f4 24 0 a 12 16 20 28

THIS MINTERH MT( 4 ) = 4 IS A CORE POINT

THE 3-IK. C UBEDCO VER I N G _ L S _ i __________ ____ — ___----- ----- ------2-CUBE = ( 27 , 10 ) = -101-THIs CUBE COVERS THESE MIN'TERMS : _ _ ...__.... . 27 10 11 26

! THIS MIN TERM MT ( . .13 ) g____ 27_____ IS A C.ORE.POINT____ -....THE FINAL CHECKING

105 105 101 101 105 205 201 101 1 0 1 101 105 j0 i c i ioi ' ...... .

MAXIMUM CUBI-I DIMENSION =

1 0 0 2 1 0THE CHARACTERISTICS t

2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

MAXIMUM CUBE DIMENSION HAS BEEN LOWERED TOJ

MAXIMUM CUBE DIMENSION HAS BEEN LOWERED TO!

ALL CORE CUBES HAVE BEEN S E A R C H E D rAND THE MAXIMUM CUBE DIMENSION IS J 1

CUBE (

CUBE (

WATCH FOR

8

80 )

10 )

HAS BEEN FOUND.

HAS BEEN FOUND.

0 AND

CHECKING POINTS 0 0

10

0 0 0 0 9999 9999

ACCUMULATED AWAYNESS

2 4

CUBE

CUBE

CUBE

f 8 14

14

1015

6 6 4 2

) » IS SELECTED.1

) HAS BEEN FOUND.

0 0

10 )CHECKING POINTS

0 0 0

HAS BEEN FOUND.

0 9999 9999 9999 9999

ACCUMULATED AWAYNESS I

8 10 10 14 r 1*

7 r

7 r 15 )

CUBE

CUBE

CUBE

j ) 5 )

8 4 2

IS SELECTED.

HAS BEEN FOUND.

HAS BEEN FOUND.

0

CHECKING POINTS 0 0 9999 9999 9999 9999 9999 9999

A29

ACCUMULATED AWAYNESS t

12 12 10 8 4 2 0 0

CUBE ( 7 r 5 ) IS S E L E C T E D ♦

CUBE ( 1 y 0 ) HAS BEEN F O U N D ♦

CUBE ( I f 5 ) HAS BEEN FOUND.

CHECKING POINTS9999 9999 9999 9999 9999 9999 9999 9999

ACCUMULATED AWAYNESS I

12 12 10 8 4 2 0 0

CUBE ( 1 , 0 ) IS SELECTED.

A 3 0

M A X I M U M C U B E DIME!'.? I o n = 3 1 .. C 0 _ 2 1 ...„_ fj_ 3... 3__ 1 _____________________

t he c h a r a c t e r i s t i c s :________ 2__5. 2_ 1_ _ 2. 2 5__L-_1_._2_JL-_2_.1_2__1 2 i._____

MAXIMUM CUTIE DIMENSION has been lowered Tg: 2

CUBE ( 1 , 4 > HAg BFEfu FOU'mD, CHECKING POINTS ... . . ...................

9999 9999 9999 9999 1 1 D 0 0ACCUMULATED Aw AYNESS :

j 0 0 0 0 2 2 4 4 4CUBE < 1 , 4 ) IS SELECTED.........

_ ALL core CUBES HAVE been SEARCHED, ________ .AND THE k‘AX I MUk‘ CURE DIMENSION IS J 2... CyEE . ( 10 , 2 ) - HAS BEEN FGuN'D.. . . . ... ... .

CUBE C 10 , 1 4 ) HAS BEEN found,i WATCH Fop f AND 14

c h e c k i n g . p o m s . <V ............. ...... ............... .9999 9999 9999 9999 i 9999 9999 q r, ACCUMULATED. AWAYNESS. : ..... ...... .....

i

J 0 C < 0 0 8 2 4 6 8

CUBE ( 10 , 6 ) IS SELECTED,CUBE <...14..., .10....).. .. HAS BEEN ...FOUND,CyBE ( 14 , 15 ) HAS BEEN fOuND,

' CHECK I N'G 'POINTS " '9999 9999 9999 9999 1 9999 9999 9999 9999

A CC UM U LA T E F T W A Y NE SS 1 '_________________0_____ C____ 0 C___ _1 Q___ 2 4 6 fi_________

CUBE ( 14 , 15 ) IS SELECTED.CUBe ( 7 , 5 ) ... HAS EEEN F O U N D , ...

CUBE____ ( 7 .... j__ 15.. 1 HAS BEEN FOUND, __________

A31

WATCH FOr< 5 AM: 15c h e c k i n g POINTS

9999 9959 9999 9999 9999 9999 9999 9999 9999a c c u m u ' t ~J 7 t x u Bss~7 “

0_____ j . '_____ J l n _______ 2 A _ 6 _____ fi________________

CUBE ( 7 , 5 ) IS SELECTED.

MAXIMUM COSE D I M E I On = 3______1....._ ... 0.... 2_____1_____L!____ 3....3... 1________________ ___ _I THE CHARiCjERlSTlCS :| 3_ i_ 3__ 1 3 5 3 1 3 5 3__5 3 1 3 1. 3 . 5 3 5 . 3 1

3 5 3 1 3 1

MAXIMUM CURE DIMENSION'. Has BEEN LOWERED TCv 2

; ALL CORE CUBES HAVE BEEN SEARCHED.! AND THE MAXI HUK ,_C wLE D I.MEN;S I ON I S : ... 2_________ ______ ...........

CUBE ( ’ 14 * 2 '*) I- A B E E hj F 0 U n D ," CUBE I 1 * , * ) HAS BEEN FOUND.I WATCH FOR ... 2 AND * _____ _____________ ____ ________________ _____

CyBE ( 1 * , B ' > HAS BEEN' FOuND,j watch fop a and ' e ~ ........

L _ CHECK INC.. PC I NTS ____________________________ _ 0 1 0 1 0 1 0 9999 1 9999 1 99991 9959

i ACCUMULATED AWAYNESS : ""........ .!_______ j 4 _2 _ _ 4___ 2 _ A 2 4 0___ 1 ____ CL_____ 2. P

' 2 0~~CUBE ( 14 . B > IS SELECTED,......................CUBE ( 1 . 7 ) HAS BEEN FOUND.CUBE ( 1 11 1 HAS bfe'n f o u n d ,

A 3 2

watch f o> CUBE (WATcH FOrCUBE (CUBE ( CUBE ( CUBE (

7 a '• D 111 , 13 )11 AND 13

1 , 7 ,

7 ,

7

13 ) 1 > 2 )

)

CHECKING } •;TS'-•■>9 9 9 9 9 ? c 9 9 9.9 " 9 7 5 9 9

ACCUMULATE.- Aw a y .vESS :42

HaS BEEN FOUND.

IS SELECTED.Has been found, has BEEN' F o u N D ,Has b een f o u n d ,

7 9 9 2 9 999^ 9999 9999 9 9 0 9 9 9 9 9 3 9999

CyBE ( CUBE (c u b e <WATCH FO

(

7

11

ll?.

CUBE WATCH FOR

U ' 2 >

1 AND . P

IS s e l e c t e d , has b ee n FOUND. has bee n f o u n d.,

has been f o u n d ’,’

CHECKING POINTS99 9 9 9 999 99 99 9 9 99 99 9 9

.ACCUMULATED AwAYNESS 1...4 6 6

2 . 0 ________________

.3 9999 9999 9999 9999 ?999 9999 9999 9999

IP

CUBE ( CUBE < CUBe ..(

11 , A

713

IS s e l e c t e d ,HAS BEEN FOUND. ...HAS BEEN. FOUND.,..

A3 3

WATCH FOR 7 AND 13’cub e < 4 » 14 > HAS bfen f o u n d .

HATCH. / O r .13 AND ..... 14_____ .____ ______ __________ _____ __... CHECKING POINTS

9 9 9 9 9 9 9 9 0 9 9 9 9 9 ?? 9959 9995 9999 9999 9999 9 999 9999 999°" 9999 9999IL_ ACCUMULATED AWAYNESS . I... ........ . ............

4 6 6 1 0 4 6 6 0 2 0 6 P, . 2 .i' ....... . ...! CUBE ( , 14 ) IS SELECTED.

m a x i m u m CvJuf d i m e n s i o n = 3 ... 1 .■ :■.... 2 .____i___ '■ ___ 3. 3 1j THE CH/j^ C iEPISt ICS :L 2 ..1 .. .1. .1__3__X__1_X.__2__1 1 1 1 1 __3..l._ 3..1 . ........

MAXIMUM fU?E DIMENSIDNyHAS BEEN LOWERED TO: 2f

CUBE ( 2 , 3 ) Has been FOUND.'’ ’... CHECKING POINTS ......................... ................... ...

1 9999 9999 0 C 0 0 0 1

.ACCUMULATED AWAYNESS.: .... ............ .| 2 0 0 4 4 4 4 4 2

CuBE < 2 V 3 ) I? SELECTED. .... ..~ ~ ~ ... CUBE ( . 5 ,.i > HAS BEEN FOUND,____________________

CHECKiNg POjNTSI__________ 9$99 9999 9999 99 99____0____ 0____ 0____ 1____ 1________

ACCUMULATED AWAYNESS : 2 O'."" 0 4 8 10 8 6 6

- cube ( 5 . ,____ i. )... is .selectedt__________ _____ __

A 34

CHECKING PCI’-'TS9999 1 1 29999 1 1 9 9 9 9 9 9 9 9 9 9 9 9 299999999 9999

ACCUMULATED AWAYNESS’”: ".... ~ ..... . ......4 6 6 4 4 6 6 0 2 0 4 02 0 ' '........

CUBE__ (_ 1 ,....13__ ) . I S SELECTED,CUBE ( 7 , i ) HAS EEEN FOUND,

j CUBE ( 7 2 ) HAS BEEN FOUND, •i1 CUBE___(.. 7.. , ...4..j.. HAS. BEEN FOUND,....................

CHECKING POINTS909? 9 ?C9 9999 3_ 9ooo__ 9999.J>?_99_..9.999 9999 99.9.9____3_ 9 0.9 059 9 9 9959

a c c u m u l a t e :; AWAYNESS :4 6 6 N A 6 6 C ? I 6 f

j CUBE ( 7 , 2 ) 1 ? SELECTED.c ube ( 11 , 1 ’) Has EEFN f o u n d ,

cube ( 11 , 2 > has been found, ... .......I WATCH FCP 1 AND 2I 1

CUBE (.. ! 1 ,.O 7 "has BEEN FOUND, ....... .. WATCH FOR 1 AND 8..........| CHECKING POINTSi _________ . 9599 .9999 .9999____3 9999 9999 9999 9999 9999 9999 9999.. 9999

9999 9999.. ACCUMULATED A way NES S :.. ..... _ _______ _______ _____ _ ______

A 6 6 10 4 6 6 0 2 0 6 0

CUBE ( 11 , 5 ) IS SELECTED.CUBE ( A , 7 ) HAS BEEN POUND.

I CUBE < 4 , .13 ) .HAS BEEN FOUND_____

A3 5

CUBE ( 1-5 15 ) HAS E&EN FOUND, CHECKING POINTS

9999 9999 9999 9999 9999 9999 0 1 gACCUMULATED' AWAYNESS 7 ""." ...... ”.

2 0 P 4 3 10 12 10 PCUBE ( 14 , 15 ) IS SELECTED.CUBE ( S , 9 ') HAS BEEN~FOUND'”CHECKING Pc I Ts

r 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 3

:__ ACCUMULATED AWAYNESS :2 C 0 a r 1 0 12 10 10

| CUBE ( 6 , 9 ) IS SELECTED.!’ ALL COPE CUBES HAVE BEEN SEARCHED,AND THE ' A X i ?■- U' CUTE DIMENSION IS : 2

c u b e < n , i ) h ;. s b e e n : f o u n d ,

CHECKING p.Op’TS *_ 9099 9999 9 99a 9999 9999 9999 9999 9999 9 9 9 9

ACCUMULATED AWAYNESS :2 C C < 10 12 10 10

CUBE < 11 , 1 ) IS SELECTED.

MAXIMUM CUBE DIMENSION' = 3r 1_ _ ..JO_. . ._2L____1____ £L___ 3 3 1 _____________

THE CHARACTERISTICS :!_________.2„... 1 2 . 1 . 2 __l 2 1 2 1 2 1 2 1 . 2 1 Z _ 1 2 .1 .. 2

2 1.. ..MAXIMUM CUBE DIMENSION Has BEEN LOWERED 1Q:........ 2I MAXIMUM CUBE DIMENSION HAS BEEN LOWERED TO: 1

ALL CORE CUREs HAVE B E E Kj SEARCHED#AND tHE M a X IM y M CyEE DIMENSION IS 5 1

A36CUBE < 31 27 ) HAS BEEN FOUND,

IL___CUBE ( 31 » ...3'J...) ..HAs BEEN FOUND. .. ...... ........ . .....

WATCH FOR 27 AND 30 I CHECKING POINTS .......'..............I o o o o o o o o n o 9999 9999i

ACCUMULATED AWAYNESS : -3 3 6 6 4 4 4 4 2 2 0 0i CUBE ( 31 , 3L ) IS SELECTED.

CUBE ( 2fi , 24 ) HAS BEEN FQUNl ,....CUBE ( 2 P , 30 } HAS..EEEN.f DL'NL,! CHECKING pGI':ts:.... o p 0 _____o. Q .. 0. 9999 _..0.. n 9999 9999 .9999

ACCUMULATE;' a w a YNESS :' 12 14 12 1? 3.0 IP 4 6 6 2 0 P

iL_.CUBE ( .2? ..., 24 ._}____IS SELECTED,....... ...... ........

CUBE ( ' 27 . 25 ') HAS b e e n FOUND,i CUBE < 27 , 31 ) HAS BEEN FOUND, CHECKING .POINTS _ _... .......

0 0 ,0 0 0 C 9999 9999 9999 9999 9999 9999... ACCUMULATED AWAYNESS :j 16 IF 18 16 18 18 4 6 6 2 C 01

CUBE (" 27 25 j fs~SEL¥cTED7......... ."'........ ...CUBE <... . 7 . ,.... 5..... ) HAS BEEN FOUND. ............. ......)

| cUBE < 7 , 6 ) HAs BEEN FOUND.WATCH F~OR F T m T 6CHECKING POINTS ........... ...... _______ _ .. . ___ ________

0 0 0 0 9999 9999 9999 9999 9999 9999 9999 9999

A37

ACCUMULATE!' AWAYNESS :?.2 '22 20 20 18 lfi 4 6 6 2 0 0

jrjgg (...' 7 ■ , 6 ) I S SE L EC TED . ~ .~ '"

cube (. s. ,___1 . ) ...has. been: fdunP.. _________________________CyBE ( 5 , 7 ) Has BEEN f Ou ND.CHECKING pOINTS

0 9999 0 9999 9999 9999 9999 9999 9999 9999 9999 9999ACCUKULATED A*a YN ESS :

24 22 24 20 3 8 18 4 6 6 2 0 0

CUBE ( 5 , 1 > IS SELECTED.* CUBE ( n ,.. 1..)...HaS BEEN F O U N D " ,

cube ( 0 , ... 2 ). .has Been rclndti CHECKING p U I N t S

_ 9 9 9 9 9 0 9 9 . 9 9 9 9 ...9,?99_ 959.9 .9999 9999 9999 ?999 .999° 9999 9999ACCUMULATED Ai-j aynESS :

........... T 24 22 2J4 2 0 18 ' 18 4 6 6 2 0 r; CUBE.. . (.. .. C . , 2 .) ... IS_SELECTED..._______________________ ____

MAXIMUM c u b e d i m e n s i o n = 4 1 CL. _j)____ 0 2____ 1____ 0 0 3 _____3____ 1_____0____ 4____ 6

4 1 T.HE— GH ATlACXE P. 1S I.I C S ;____________________________________________________ -1 1 1 1 2 1 2 1 3 5 3 1 3 1 3 5 3 5 3 1 3 1

3 5 3 5 3 5 3 5 3 5

j MAX IMyM CyBE DI MENS I ON HAS 'BEEN LOWERED Tq: " 3 CUB-E_ A 6... „___24___)___ HAS..BEEN FOUND. - ...

A38

c h e c k i n g p o i n t s9 9 9 9 O l d 9 9 9 9 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999

.A C C U HULA IE D AWAYNESS ;0 4 0

CUBEc u b e

(

(

n 24 ) IS SELECTED.3" ) HAS EEEN POUND,

CHECKING P ^ r ’Ts5 9 9 9 9 9 9 9 n 9 9 9 9 9999 9099 9999 9999 999Q 9999 99909999 9999 9999 9999

a c c u m u l a t e ;: a w a y n e s s :4D

CUBEc u b e

141 6 ,

6c

3L )

2 a >

12r. 0

is s e l e c t e d ., has EEEN FOUND,

c h e c k i n g p o i n t sf9999 9999 999% r. 9 9 9 9 9999 9999 9999 9999 9999 9999 99999 9 9 9 9999 9999 9999

ACCUMULATED awaynESS :80

160

CUBECUBE (

16 1 9

28 ) IS SELECTED,31 ) HAS BEEN FOUND,

c h e c k i n g p o i n t s___________ 9 9 9 9 9999 99 9 9 9 9 9 9 9999 9999 9999 9999 9999 9999 9999 9999999 9 99 9 9 99 99 9 999

ACCUMULATED AWAYNESS..:.8 160 0

c u b e < 19 , 3i > is s e l e c t e d ,

A 3 9

all c o r e CUBES HAVE h efn SEARCHED,' and THE MAXIMUM CUBE DIMENSION IS :MAXIMUM CUBE DIMENSION = 4

0 0 0 2 1 0_1.4

,THE... C H A R a C-7.E RIS.j LC_S...:_____________________________3 i 2 l 3 l 3 5 3 5 3 l 2 i3 1 3 5 3 1 3 , 1 2 1 1 1

MAXIMUM CUBE DIMENSION HAS BEEN LOWERED Tq: _CUBE___ L 3 . , 6__)..._... H.A,s £EEn .FPUnP,..CHECKING POINTS

9999 2 9999 99999999 9999 9999 9999 9999

ACCUMULATED AWAYNESS : 2 0

CUBE ( 3 , 6 ) IS SELECTED.MAXIMUM CUBE DIMENSION H/S BEEN'LOWERED TO:

1

CUBE < 26 , 2< ) H a s B E E N E n U a, D ,CHECKING POINTS _ j, 9999 2 9999 9999 _ 2.. 0

9999 9999 9999 9999 9999ACCUMULATED AWAYNESS.:

10

CUBECUBE'

262 6

24 )

CHECKING POINTS.

IS SELECTED.HAS EEEN FOUND',

2 9999 3 9999 99999999 9999 9999 9999 9999

ACCUMULATED AWAYNESS :_ 8 0___ 10o' 6 o

. 12

2 99 9 9

r, 0 9 0 9 0 9 9 c

I 9999 9999

2 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

14___ 14

CUBE... ( 28 , 6 ) IS SELECTED.a ll c o r e CURES HAVE BEEN SEARCHED,AND THE HAX I MUN _ CUBE _D I MENS I ON I S :_____2CUBE < c , 6 ) HAS BEEN FOUnjD,

r.CUBE ( o , 12 ) Has BEEN found,"I CHECKING Pon-TS ....... . .. __

C099 9999 9999 9 9 99 9999 2 0 2 9999 9 999 9999 99999 99 9 9999 999 9. 9999 9999

ACCUMULATED AWAYNESS :8 0 10 U 0 16 18 16 4 6 0 0c fj 0 n c

I CUBE .... L C , 6 ____IS SEi.FCtED,__________ .____ ____ __________! CUBE ( 15 , 6 ) HAS BEEN FOUND,

CUBE < 15 23 ) HAS b e e n FOUND,WATCH FOR 6 AND 23

i CHECKING PC I NTSL _ p 9 99 9599 9959? 9999 9999 .9999 0....3__99_9_9._9.999 999.9..9999

9999 9999 9959 9999 9999.. . ACCUMULATE! A WA YN'ESS :j B 0 1 0 0 0 1 6 22 20 4 8 0 0! . C Q ... . 0.. . 0 .. 0. . . „ _______________ ___

CUBE < 15 6 ) IS SELECTED." ' CUBE < 22 , 15..)...HAS BEEN FOUND,....:_ CHECKING . POINTS _______ ______________ _______________ ____________________

9999 9999 9999 9999 9999 9999 0 9999 9999 9999 9999 99995999 9999 9999 9999 9959

ACCUMULATED AWAYNESS :. . e 0 . 10 0 Q _ 16 24 20 .4____ 8____ 0... 0

0 c 0 0 0. CUBE (. 23 , 15 ) IS SELECTED.

A41

CUBE ( 21

CUBE < 21.WATCH FOR . 23

23 ) HAS BEEN FOUND,17 ) HAS BFEN FOUND,

17 .. ___ _________CHECKING POINTS 9999 9999 .9999. 9.999 9999 9999_9999 9999 .9999 9999 9999 9oco

9999 9999 0999 9999 9959..ACCUMULATED A w n".E SS..

fi C

CUBE (

100

21

16 24 20

17 ) IS SELECTED

.____________

MAXIMUM cur-E DIMENSION = 3 1..... ■;... .c... 2 1 jt he CHARACTERISTICS : r _3__ ?__3 1 2 1 *3 5 3_5 3_ 1_..3 _1__3___1__ 3 . 1 3.. 5

3 1 2 1 3 1

CUBE ( 3 , 10 ) HAS BEEN' FOUND,CHECKING PC I NTS_ 2 9999 9999 _ _ 0_ _ 2 9999_9999____ 0 0 0 0

1 0...ACCUMULATED. a Wa YNESS :.

2 0_.. .2. 6.

0

3 ,CUBE (CUBE ( CHECKING PCI’vTS

10 ) IS SELECTED.24 " ) ' HAS BEEN FOUND,"

9999 9999' 99 9 9' 9999 9999" 9999 ' 9999 9999 9999 9999 9999 1 9999

1

A42

u *■' r ■'• ■.-'ACCUMULATED AWAYNESS :

2 . .._. D 06 6

CUBE ( A , 24 )__ IS SELECTED....CUBE ( 27 , 10 ) HAS BEEN FOUND.CHECKING POI'-TS

9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 999Q 9999 9999.............. ....... ................... ...ACCUMULATED AwAYNESS :

2 0 0 4 2 06 6

CUBE < 2 7 , 10 ) IS SELECTED.ALL CORE CUBES HAVE BEEN SEARCHED,AND THE A X I U S' CURE DIMENSION IS : 3

1! * # p 0 R 7 B A N ■»# Sr0 P!__/LOGOFF _ _________ _________

C E420 LOtOFF AT 231k C*i U / 3 0 FOR TSN 239 0 , C E421 Cp,j t IME j SEDV 00 0090,4681 SECONDS.

APPENDIX C

EXAMPLE OF 6 VARIABLES, 18 MINTERMS

A44

toatal number or on-set and dont«cares =ON-SET = 18 pONT-CARES = 0

NUMBERS OR wAR j ABL£ S_ =_ 6_______

MINT.# 1 = 6MINT.# 4 = 9MINT.# 7 = 27MINT.# 10 = 28M INT.# 13 = _ 19MINT.# 16 = 0

MINT,# 2 MINT.# 5MINT,# 8 MINT,# 11 MINT,# 14_ MINT,# 17

18

A MINT,# 3.11 MINT,# 630 MINT,# 916 MINt.# 1261 MINT.# 1544 MINT.# 18

B1429174536

THE c u b e c o v e r i n g LIST :

THE 1-TH CUBE COVERING IS f ____ 1-CUBE... = (_36 , 44 ) = ... 10-100THIS CyBE COVERS j hESE MIN'tERMS j_4 A 3.6_______________________________ ___________

•____i ....... i...... ..... ..... ..... .....1

THE 2-TH CUBE COVERING IS \________1 - C U_B E_s..( 6 1 __!_•_=______1-11.01THIS CUBE COVERS THESE MIN'TERMS s------ 6.1-— 1 5 -------- J----------------------------

THE 3-TH CUBE COVERING IS I _1-CU3E = ( 26 , 29 ) = 01110-

THIS CUBE COVERS THESE MJNTERMS |________2S Z&_________________________________________

THE 4-TH CUBE COVERING IS I ________L-C41BE...g (___ 4 1 Q ) s______ Q.e_0-0.0

THIS CUBE COVERS THESE MINTERMS t________4___ .0______________________ ________

A45

--- — ■ ............THE 5-TH CUBE COVERING IS 1

l-*CUgE = ( 8 » 9 ) s ooioo-

THIS CUBE COVERS THESE MJNTERMS » A 9 ......

THE 6-TH CUBE COVERING IS 1 1-CURE = ( 14 # 30 . ) s o - m o ..... ... .....

THIS CUBE COVERS THESE MJNTERMS | 14 30. .......... .........

THE 7-TH c u b e c o v e r i n g IS I1-nUBE = ( 6 * 4 ) = 0001-0 ................ .

THIS CUBE COVERS THESE MJNTERMS 1 6 4

THE 1 fi-TH CUBE COVERING IS 1 1-CUBE = ( 16 , 17 ) = 01000-

THIS CUBE COVERS THESE MJNTERMS 1.16 17______________ ____ ______

«

THE 9-TH CUBE COVERING IS 1 ..1-CUBE _= ( 11 *..2 7._> s ____ 0-1011_____ _____ __ ____ ______

THIS CUBE COVERS THESE MJNTERMS j .11 .27 . __________

the 10-TH CUBE COVERING IS 1 1-CURE = ( 19 , 17 ) z 01D0-1

THIS CUBE COVERS THESE MINTERMS » 17 19

THE FINAL CHECKING____________ 105 205 105__ 105 105 105 105 105 105__105__i 05 20

105 105 105 105 105 105

A 46

rprry

MAXIMUM CUBE DIMENSION

0 0I4

01

THE CHARACTERISTICS 2 1 3 1 22 1 2 1 2

2o

1.1.

23

X) O"

2o

’■yn

MAXIMUM CUBE DIMENSION HAS BEEN LOWERED TOJ

MAXIMUM CUBE DIMENSION HAS 'BEEN' LOWERED TO?

MAXIMUM CUBE DIMENSION HAS BEEN LOWERED TOJ

ALL CORE CUBES HAVE BEEN S E A R C H E D ? AND THE MAXIMUM CUBE DIMENSION IS 1CUBE ( 36 , 4

CUBE < 36 , 4 4

WATCH FOR 4 AND

CHECKING POINTS0 00 0

t

) HAS BEEN F O U N D .

) HAS ' BEEN 'FOTJNBT

44

00

0 0 0 0 9999 9999”

1

1 2

0 0

'V

0

4 10

CUBE

CUBE

CUBE

36

24

44

614 4

4 64

8' 0

40

IS S E L E C T E D ♦

HAS” BEEN FOUND,

HAS BEEN FOUND,

CHECKING POINTS 0 0 0 0 9999 9999

0 00

0 9999 9999

ACCUMULATED AWAYNESS

1218

CUBE

CUBE

CUBE

45

2828

84

6130':>9

10o

1012

14 104

HAS BEEN FOUND.

10

0

16

0 0

12

IS SELECTED.

HAS” B E E N ” FOI.J N D 7

0

8

0

14

8

0

14

A47

WATCH FOR 30 AND

CHECKING POINTS

29

0 0 0 9999

09999

0 0 0 0 9999 9999

0 0 9999 9999 0 0

ACCUMULATED AWAYNESS t

18 1 2 24 6

146

14 2 0 14 18 6 1 0

2 0 """ 14 8 8 18 18

CUBE ( 28 r 29 ) IS S E L E C T E D ♦...

CUBE ( 4 r 6 ) HAS BEEN FOUND,

CUBE ( 4 f 0 ) HAS BEEN FOUND,

WATCH EOF 6 AND o ....

CUBE ( 4 v 36 ) HAS BEEN FOUND.

CHECKING POINTS0 9999 0 9999

0o p o o 0 0 0 9 9 9 9 9 9 9 9 • — 0 0 9999 9999 0 0

ACCUMULATED AWAYNESS •>

* 2 0 1 2 30 14

161 2

* 18 26 18 ’ 18 1 0 1 2

28 2 0 14 1 2 2 0 2 2

CUBE ( 4 r 0 ) IS SELECTED.

CUBE ( 8 t 9 ) HAS BEEN FOUNBT' ...

CUBE ( 8 ? 0 ) HAS BEEN FOUND.

CHECKING POINTS0 9999 9999 9999 0 0 0 0 9999 9999 0 0

0 9999 9999 9999”999Sr ~9999~

ACCUMULATED AWAYNESS t

36CUBE ( 8 ycuBt TCUBE ( 14 v

1620

1616

.1.820

2814

2218

32

9 ) IS S E L E C T E D ♦

- 3T - 5----- FI A T n i E E N ~ T O U N D .'

30 ) HAS BEEN FOUND.

26 18 16 24 26

WATCH FOR 6 AND 30

A48

'c h e c k i n g p o i n t s --- ----.•»■**«> a. -■: W ..- • ;• \ f •■■■■ i- -

0 9999 0 9999

99999999

9999 0 9999 9999 9999 9999

0 9999 9999 9999 0 0

ACCUMUI..ATED AWAYNESS J

28 2 0

42 262 02 2

' 24 '32 ---2226 18 24

3 6 ...26 18 30 34

CUBE ( 14 f 30 ) is s e l e c t e d ;---- - - -

CUBE ( 6 ? 4 ) HAS BEEN F O U N D ♦

CUBE ( 6 y 14 ) HAS BEEN FOUND.

WATCH FOR 4 AND 14 ------------ ----- -----

CHECKING POINTS< p n c p < p 9 9 9 9 9999 9999 0 9999' •' 0 9999 9999 9999 0 0

0 9999 9999 (p 9 cp 9 9 9 9 9 9 9 9 9

H L L U n U1... r-i 1 1- .Li R W f-t I N L S S ;

28 2 0 24 30 38 24 4 4 30 28 p p 34 4048 34 28 9 9 p p 2 6

CUBE ( 6 s’ 4 ) IS SELECTED.

CUBE ( 16 1 , 17 ) H/fs BEEN FOUND.

CUBE ( 16 r 0 ) HAS BEEN FOUND. — - -

CHECKING POINTS9 9 9 9 9 9 9 9 9999 9999 ~0""9999 " 0 '9999 9999 9999 9999 9999

0 9999 9999 9999 9999 9999

ACCUMUL ATED AWAYNESS ♦♦ --- ------------------ ----- - -■ - .

34 24 28 34 44 32 48 36 32 26 34 4050 40 36 30 30 32

CUBE ( 16 s’ 17 ) IS SELECTED.

CUBE

CUBE

< 11 S’ 9 ) HAS BEEN FOUND.

( H r 27 ) HAS “BEER POUND.

CHECKING POINTS >9999 9999 9999 9999 9999 9999 9999 '9999' 99 9 9 9999 9999 9999

0 9999 9999 9999 9999 9999

ACCIJMUI..ATE[D AWAYNESS ♦ " .......

40 32 32 3 6> 44 36 4852 46 42 36 ' 3 8 .“ “42

CUBE ( 1 1 r 27 ) IS SELECTED.

CUBE ( 19 y 27 ) HAS BEEN FOUND.

CUBE ( 19 y 17 > HAS BEEN FOUND 7

WATCH F OR 27 AND 17

CHECKING P01 NTS

40 36 32

A49

• ;T :);,j

40 44

999? 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9 9 9 9 9999 9999 9999 9999 "9999

ACCUMULATED AWAYNESS J

46 38 3850

4040

4848

4450

50 46 40 38 42 44

CUBE ( 19 17 ) IS S E L E C T E D .

f V1

APPENDIX D

EXAMPLE OF 5 VARIABLES, 20 MINTERMS

A51

TOATAL NUMBER OF ON-SET AND DCN'T*CARES = 20ON-SET = 20 DONTrCARES = 0NUHBERS OF VARIABLES = 5 ____________

MINT.# I B 1 MINT,# 2 = 3 MINT.# 3 = 5-HI N T , #____4__ -_____ 4_____ MINT.# 5 = 6 HINf.# A =____9MINT.# 7 s 11 MINT,# 8 s 10 HINf.# 9 = 25MINT,# 10 s 27 MINT,# 11 = 26 MINT,# 12 - 30.M IN T, #. 13. ..=____2 9___ M INT.# _% 4 ,__s 2 8 _MJ.NI. ,# 15 = 19MINT,# 16 = 0 MINT,# 17 = 18 MINT,# IB s 22MINT.# 19 = 21 MINT.# 20 = 20 MINT.#

THE CUEE COVERING LIST j *

THE 1-tH CUBE C0VERING IS 1 . 2-CUBE s ( 20. ,__29 1-10- _ ..................

THIS CUBE COVERS THESE m INTER^S 1 29 ?fi 21 20

f '1

THE 2-TH CUBE COVERING IS 1 2-ClJpE = ( 18 , 30 ) = 1 — 10

THIS CUBE COVERS THESE MJNTERMS | 2 a 30 la 22

THE 3-TH CUBE COVERING IS » 2-cUBE = ( 0 * 5 ) = oo-o-

THIS CUBE COVERS THESE MJNTERMS 1 1 b 4 0

the 4-TH CUBE COVERING IS f 2-CURE = ( 11 . 19 1 =

\

— Oil

THIS CUBE COVERS THESE MINTERMS i 3 11 27 19

A52

THE 5-TH CUBE COVERING IS I 2-C U BJL J=_ (___ B , 11__ ) = 010THIS CUBE COVERS THESE MJNTERMS | ________6___ 9__11__10_______________

THE 6-TH CUBE COVERING IS I2-CUpE = ( 25 , 11 ) s _____*1Q~1_______________________________

THIS CUBE COVERS THESE MJNTERMS i________9 11 25 27________________________:_______________

THE FINAL CHECKING_U 5 1P 5__ 1_0 5 10 5 115 205 305 105 105 205__1D5 105

105 105 105 105 105 105 105 105

A53

***

118

■lJ21

4209

T'i~ T O 27 26 30 29 28

MAXIMUM CUBE DIMENSION =

1 0 0 0 24 1

THE CHARACTERISTICS J4 1 3 1 3 1 3 3 1 3 1 3 1 3 ' :L

33

0

4 1“3 1

5 3I 3

33

MAXIMUM CUBE DIMENSION HAS BEEN LOWERED TOi

MAXIMUM CUBE DIMENSION HAS BEEN LOWERED TO?

ALL CORE CUBES HAVE BEEN SEARCHED?AND THE MAXIMUM CUBE

CUBE ( 20 y 5

CUBE ( 20 y 30

WATCH FOR b AND

CUBE ( 20 1 y 29

WATCH FOR 30 AND

HAS "BEETTTOUND .

HAS BEEN FOUND.

30

HA6 BEEN' FOUND?f

CHECKING POINTS0 0

9999 9999

ACCUMULATED AWAYNESS

29

10

10

0'O'

0 0 0 1" 9999 9999

0 4 6

4 1 4 1

9999 9999 0 9999 9999 9999 9999

0 0 1

40

60

2 2 4 4 4 4 4 2

60

60

2 4 4

4CUBE ( 2 0 y 29 ) IS SELECTED.

CUBE ( 18 y 27 ) HAS BEEN FOUND.

CUBE ( 18 y 30 ) HAS BEEN FOUND.

WATCH FOR 27 AND 30

CHECKING POINTS 0 0 1 1 0 0 0 1 1 1 9999 9999

CUBE

CUBE

CUBE

WATCH FOE

!D AWAYNESS *♦

1 0 1 0 8 3 8 1 0 1 0 S . 6 “ 15---- A4 2 3 8 4 2 4 2 4

18 , 30 ) IS SEL ECTED. -- ■ • ■

0 y 5 ) HAS BE EN FOUND,

0 y 9 ) HAS BE EN FOUND.

5 AND 9

01 NTS9 o 9 9 1 9999 9999 " 1 r 0 1 ... 1 1 9999 999S>999 9 9 9 9

ACCUMULATED AWAYNESS i

1 9999 9999 9999 9999 9999

108

CUBE ( 0 i

CUBE ( 11 !

CUBE ( 11 iI

CUBE ( 11 v

WATCH FOE

CUBE ( 11 r

CUBE ( 11 r

WATCH FOE 25

126

810 8

108

123

146'

124

)1 )

)8 AND

t 2 6 ,)

y 19 )

IS SELECTED,

HAS BEEN FOUND,

HAS BEE?TTT)UNDT

HAS BEEN FOUND,

HAS BEEN F O U N D ,

HAS BEEN FOUND,

AND 19CHECKING POINTS

9999 9999 9999 9999 1 2 99999999 9999 9999 9999 9999 9999 9999

ACCUMULATED AWAYNESS ?

1212

CUBE

CUBE

CUBE

1188

12 1219 )

1 )11 )

1210

1 2 ---14“12 10

1 110

“14“10

D9999

1 410

IS SELECTED.----

HAS BEEN FOUND.

HAS BEEN FOUND.

10 12 10 £

2 9999 9999 9999

12 12 12 i:

A55

c h t l; k j. n i.7 I-' o ;i. n i s9999 9999 9999 9999 9999 9999 9999 9999 3 9 9 9 9 9 9 9 9 9 9 9 99999 9999 9999 9999 9999 9999 9999 9999

ACCUMULATED AWAYNESS i

14 14 16 16----14 14 14 1 4 ” "14 1416 16 14 14 14 16 16 16

CUBE ( 8 , 1 1 ) IS S E L E C T E D , - -----

CUBE- < 25 , 1 1 ) HAS BEEN F O U N D «

CHECKING POINTS9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999""”9999 ~9999 ” “ “

ACCUMULATED AWAYNESS :

16 16 20 22 16 14 14 16 14 14 16 20IS 20 16 18 18 22 20 22

CUBE _ <___25 y 11 ) IS SELECTED «

i1

A56

APPENDIX E

EXAMPLE OF 7 VARIABLES, 53 MINTERMS

A5 7

T 0 A T A L N u M E R 0 r c-> - 5 E ' A N D D O N T * C A R E S = 6 3O N - S E T = 5 3 i;nu * - C A R F S = i cn u m b e r s O F V A R ! / ai-F 5 = 7

M I N T , # 1 = 5 H I N T , # 2 1 0 M N T , # 3m i n t . # 4 - 1 4 m i n t .# 5 r 2 1 M N T . # 6 SM I N T . # 7 - 2 9 H I N T . # “ .. a 31) H N T . # 9 =m i n t .# 1 0 - 3 7 H I N T . # 11 = 4 2 M N T . # 1 2M I N T .# 1 3 = 5 0 M I N T . # 1 4 = 5 3 M N T . # 1 5 =H I N T . # 1 6 = 6 4 H I N T . # 1 7 = 6 5 M N T . # 1 8 =H I N T . # 1 9 j; 6 7 M I N T . # 2 0 = 6 8 M N T .# 2 1 -M I N T . # 2 2 = 7 0 H I N T . # 2 3 r 7 1 M N T , # 2 4 =m I N T . # 2 5 - 7 4 ^ i k T . n " " 2 6“ ' r: 6 U M N T . # 2 7 zH I N T . # 2 8 - 8 2 H I N T , # 2 9 - 8 3 M N T . # 3 0 z

M I N T . # 3 1 = 8 5 M I N T . # 3 2 8 6 M N T . # 3 3 -

H I N T . # 3 4 = 8 8 M I N T . # 3 5 - 8 9 M N T . # 3 6 =M I N T .# 3 7 = 9 1 M I ,\ T . # 3 8 = 9 2 M N T . # 3 9 =h i n t .# 4 = 9 4 M J \ T . # 4 1 r 9 5 M N T . # 4 2 =M I N T • # 4 3 = 9 7 I n t . * ' 4 4 - " 98 H N T . # 4 5 =M I N T . # 4 6 = 1 6 0 H I N T . # 4 7 1 0 1 M N T , # 4 8 rM I N T .# 4 9 = I U 3 M I N T . # 5 0 1 0 4 H N T . # 51 =

. . . . . . . H I N T .# 5 2 s 1 M I N T . # 5 3 1 0 7 " H N T . #' 5 4 ' sM I N T . # 5 5 = 1 u 9 HI N'T.# 5 6 r 1 1 0 M N T . # 5 7 =M I N j . # 5 3 = 1 1 2 H I N'T . # 5 9 =: 1 1 3 H n t .# 6 0 =M I N T . # 6 1 = 1 1 5 " Y i n T . fl­‘ 6 2 1 1 6 - K N T . # 6 3 =

. ... -.

1 3_2234 4 5 6 1 66 6 9

_ 72ei 84 R 7 9 C 9 3 9 6 "99 102 1 0 510®1111 1 41 1 7

/T H E C'JBc C O V E R I N G l I S ' ''

THE 1-TH CUBE COVERING IS :3-^UBE = ( 13 > 5 3 ) = 0--- 101THIS CUBE COVERS THESE MJNTERMS : _ 13__ 5 3 ___ 5__ 21__ 2_?__ 37 4 5 __ 6 1 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _T H I S M I N T E R M M T ( 3 ) = 1 3 I S A C O R E P O I N T

THE 2-TH CUBE COVERING IS I2-CUBE = ( 22 , 94 ) = - 0 1 - 1 1 0

T H I S " C u B E COVERS' j H E S E' I K j E R M S ' :22 9 4 3 0 86

THIS M INTERN M T ( 6 22 ' f S ~ A~C C R E~P07nT

A58

THE 3-TH CUBE COVERING, i s. «...........2-C J i: E = ( 5 L , 98 ) = -1-UOlO

THIS CuBE COVERS THESE MINTERMS : 50 96 34 114THIS M I N T E RM MT ( 13 ) s 5 0 I S A C C R E P_0 I N T

THE 4-TH CUBE COVERING IS I4-CUBE r ( 89 > 06 ) = 101----This cube covers These mjntermS 569 66 60 81 82 83 84 85 87 80 90 91 92 93 94 95THIS MINTERM MT( 35 > = 89 IS A CC-RE POINT

THE 5-TV cube c o v e r i n g IS 54-CUBE = { 105 » 102 ) =------- 110--

THIS " cT7Bf'“cT v fRS~~TH£5E INTER? S : ....... ............10 5 i u 2 96 97 98 99 ICO 3.0 I 103 104 106 107 1 08 109 110 111

THIS MINHERM MT( 51 V = 105 IS A CORE POINT

THE 6-TH CuB_E C0 VE_R 1 _N_G_ _I_S_ »____________ _ __________________4 - C U B E Y { 6 7 '/ 0 4 ) s _ 10 - --- "

THIS CUBp COVERS THESE >‘INTERyS :6 7 6,4 6 4 6 5 6 6 68 69 70 71 80 81 82 83 85 86 8 7

THE 7-TH CUBE COVERING IS J4 - c U 3 E = ( 117““7 6 ‘4 ) = 1 — U - 0 -

THIS CUBE COVERS THESE MINTERMS :117 64 6 5 68 69 80 81 84 85 96 97 100 i 01 112 il.3 116

THE 8-TH CUBE COVERING IS I' 4-cUgE = < 114 , 65 ) = 1-- 0 0 --i! THIS CUBE COVERS THESE MJNTERMS :

1 1 4 65 64 6 "6 67" ' Bo " 01 02 83" 9 6 97 98 ' 99 H ? n 3 ll5

A59

l THE q-TH CJBE CCVf RING IS i^ CUBE“ = ' ( 7Y~'t " 8 2"' "> ' = """ 10 - - 0 - 0 “

THIS CUBE COVERS THESE MINTEPHS :72 “ 62 6^ "66 74 80 88 90........

THE lu-TH CUBE COVERING IS S2-CUBE = ( 10 » 106 ) = — 01010

THIS CUBE COVERS THESE MNTERvS :ref i'o6~ < 2~ ' 7 4 "........... f

THE 1 1 - T H CUBE COVERING IS i1-CUBE = ( 1 A i 30 ) r 00- 1 n o

THIS CUBE COVER S THESE wINTERhS j1 a 3.

THE F I N A L CHECKING1 0 5 10 5 1J1 ' 10 5 1 fi 1 101 101 20i 105 101 105 1011 0 3 101 101 405 3 05 305 205 205 205 105 10 5 1052 0 5 505 405 405 305 305 305 205 205 205 101 205

.. + 101" "101 10(5 105 101 305 305 20 5 205 205 205 105105 10 5 1b1 205 101 209 1 05 209 209 205 205 1052 09 2 0 9 10 5

--... ..- ---------- ......_ _ . ----- --

max I ’-'UK nuf- E DlyEN'Sl'JN = 5

THE CHARACTERISTICS :4 1 3 1 3 1 2 1 4 1 2 1 4 1 3 1 3 1 4 1 3 14 1 2 1 4 1 3 1 5 5 5 1 5 5 5 1 5 1 5 5 5 1r; 1 4 1 5 1 5 5 K 5 5 5 5 5 5 5 5 5 5 5 5 1

r ....... 5 1 ' 4 1 5 1 4 1 4 1 5 1 b 1 4 1 5 5 5 5 e; 51 5 5 5 5 5 5 5 1 5 1 5 1 4 1 5 5 4 1 4 1 5 11 4 1 4 1w- 5 1 5 1 5 1 4 1 4 1 5 1

CUBE < 13 9 53 ) HAS BEE?' FOUND.CHECKING POINTS

9999 0 9999 0 9 9 9 9 0 9999 0 n 9 9 9 9 0 9999, 9 9 9 9 9 9 9 9 0 2 n 0 2 e 0 2 n: ~ r . . . . 2 t) 0j 2 e Q 2

....... .._ ...H1 8 n 1 0 2 0 n 2 8 0

2 0 1 n r. 9999 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9999 999c9 9 9 9 9 9 9 9 0999

ACCUMULATED AWAYNESS :6 0 4 0 4 0 4 6 n 6 r

6 4 4 e 6 4 r. 6 4 6-... ............................... ..................— 8 .... ■■. ft 4 ■ V ■ 6 . — 4 ~2 ' 6 4 • 6 4 . 8

f 6 44 £ 6 4 6 4 8 6 4 2 ft6 * 4 8 6 0 0 0 0 n 0 P

CUBE ( 13 , 53 > IS SELECTED.CUBE ( 22 , 94 )* HAS BEEN FOUND.C H E C K I N G P d N T S

9 9 9 9 c 9 9 9 9 9 9 9 9 9 9 9 9 5 9 9 9 9 9 9 9 0 9 9 9 9 0 9 9 9 0

; 9 9 9 9 9 9 9 9 2 2 0 4 8 4 4 0

■' . ' “ . . . . . . . . . . 2 ? 2 -- 4 •. “ 2 .. 6 1 0 9 9 9 9 ft ... - . 1 4

1 5 1 0 9 9 9 9 5 0 2 0 0 2 8 ?2 0 1 0 C 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

... . .. 0 9 9 0 9 9 9 9 9 9 9A C C U M U L A T E D A W A Y N E S S •

D 1 0 0 6 0 4 0 4 1 2 0 1 2 n1 0 0 0 1 2 1 2 1 2 1 2 6 8 8 8 1 212 10 ... ITr 1 0 1 0 6 6 6~~ ft 1 0 1 u 1 fl1 6 6 6 6 1 4 1 4 1 4 1 4 1 C 1 0 1 0l:: 14 14 14 1 4 0 0 0 0 0 0 0

A61

CUBE ( 2k . 94 ) IS SELECTED.CUBE ( 5.i , 98 > HAS BEEN FOUND *

' c h e c k i n g p o i n t s ..... .j 9999 C 9999

9999 9999 999902

9999 9999 2 6

9999£

99994

9999P.

99996

14

9999n

4 4 2 1 5 10 4 2 1

9999 9999 9999

r9 999

4...65 4 1 9999

1 0 4

9999999999999999

64

999924

999918

999966

9999

ACCUMULATEp AWAYNESS :o 14 r-

1 ■: U n1 7

160 4

18 140

164

141216

n12

1414

0IP

16 14 16 16 14 16 14 16 2 0

124

16.14 12 14 16 13 0

T A ~

180

6140

1?160

16140

1816a

1412n

p [i r.

CUBE ( 5\ . 9 6 ) IS Ke l e c t e d .MAXIMUM CUBE DIMENSION Ka S BEEN LOWERED l C : 4CUBE ( 69 . 66 ) HAg 3 E E n, F Q U (xj D 1

CHECKING fPOI NTS 1 9999 0 9999 9999 9999 9999

6 9 9 9 9 9 9 9 906

99999999 9999

6 10 9999 9599

99956

99999999

89999

999912

99909999

10999Q

18

99999999

29999

9999 9999 9999 5 3 2

9999 9999 99999999c 9999 5

2 99995

99 9999999999

59999

59999

99999

79999

ACCUMULATED AWAYNESS :16 0

10 0 0 18 14 16

16181?

0 4 20 16 14 12

01814

4166

1?1812

n1416

201618

02014

16 14 16 18 22 240 0 0

620

14 20 22 0

‘ 22 0

140

200

180

200

16n

CUBE < 89 . 86 ) IS SELECTED.C U B E ( 105 , 102 ) HAS BEEN FOUND.

A 6 2

C H E C K I N G p o i n t s9 V99 6 999? 0 99 9 9 9 9 99 99 99 999 9 9 9 59 9999 2 999 09 5 9 9 9 9 9 9 9 9 9 9 10 10 14 10 12 16 14 12 4

12 9999 9999 999s 9999 9999 9999 9999 9999 9999 ^999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99995599 9999 0999 9999 9999 9999 9999 9999 9999 9999 9999 90999999 9999 9999

ACCUMULATED AWAYNESS :I 22 fl 2 0 0 4 0 4 12 0 22 0i 0 0 2 0 22 18 20 18 20 16 19 2?

20 14 16 1? 14 12 14 6 12 16 18 1416 4 16 ' A ' 14"" ‘20 .'22'~ ' 14 ' 2n 13"' 20. 1615 22 24 20 22 0 0 0 0 0 0 n

0 0CUBE < 105 . 102 ) IS SELECTED.ALL C 0"E E~ C U E E S Ti A V E D E E S E A fi C H E D ,......AND tHE MAXI MUV CUBE DIMENSION IS V 4CUBE ( o7 j 84 ) pAs BEEN FOU,\jD.CUBE ( 6 7 , 100 j HAS BEEN FOUND.WATChEOB 8 4 A\D 10 0

, rCUBE ( 67 , 112 )' HAS BEEN FOUND.CHECKING POINTS

999? p 9955 6 9999 9999 9999 9999 9999 9999 2 9999990? 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 6

16 9999 999? 9959 9999 9999 9999 9999 9999 9999 9999 9990" 9 99 9 9 99 9 5999 9 99 5 999 9 9999' 999 9 9999"999'9 ' 9999 '9999 99999999 9995 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999

ACCUMULATED AWAYNESS :0 26 0 24 0 4 0 4 12 0 28 n1 1 0 O 20 22 18 20 18 20 16 1 8 24

22 14 16 1? 14 12 14 6 12 16 1 8 3 416 14 . 1 6 - - 6 “ 14 20 22 14 " 20 18 20 1618 22

O240

2 0 22 0 0 0 0 0 0 n

CUBE ( 67 , 84 ) IS SELECTED.CUBE- < 117 ;..“64 j HAS BEEN FOUND.

A 6 3

CHECKING POINTS9959 0 9999 r. 9999 9999 9959 9999 999q 9999 2 99999999 9999 9959 9999 9959 9999 9999 9999 9999 9999 9999 p

16 9999 9999 9999 9959 9999 9999 9599 9999 9999 9999 99999 99 9 9 9 9 9 9 99 9 9 99 <f~99"9'9~9 999~~5995 '“99 99 9 95"o"9 99q~ QQ9 9"9999'9999 9999 9999 9999 9999 9999 9999 9999 9999 5 999 9999 9999 9999 9999 99 9 5

ACCUMULATED AWAYNESS :

1ii

1 ' 2 6 16 18ij

32n1414220

C1616240

3020126

20

0 22 1 4 14 22

... AIB12200

Q_201422.0

-4-166

14‘0

122n12

..20fi

01616180

_ •1818200

0 26 3 4 16 0

CUBE < 117 , 64 ) IS selecte D ,CUBE ( 114 65 H A S g E E N f o u n d •CHECKING POINTS

-> 0 95'' 9 9 99 9 9 9 9 9999 9999 99 9 9ff

9 9O99990

18 99 9 9 9999 9999

C999 9 9999 999 9 99 9 9 9999

r\

9 9 9 r 9999 9999 9999

"95 99 9999 9999 9999 9999

95 99'9999999999999999

99999999999999999999

99999999999999999999

9999995Q999999999999

9999 5 999 9 999 9999 9999

" 2 9999 9999 9999 9999

9999in

9 9 9 99 0 5 99999

1ACCUMULATED ArfAY UE SS

1

1 ; 26

— 3 6 ' U

1400

16.■ 3 6

26 12

02214

41012

0 20 ■ 14

..4186

122012

01616

381818

r>2814

1 6 18

1422

1624

"" 5" 2 0

- j 4- 22

200."22

0..14 "

020"0

"18n- 20

016n

0 0| CUBE ( 114 65 ) IS SELECTED,I ‘ CUBE ( 72 I 82 > "HAB^BEE^TOTA’D". “ “

CUBE ( 72 , 9 8 5 HAS BFE* FOUNDWATCH F OH 82 AND 98CHECKING POINTS

9999 1 9999 0 9999 9999 9999 9999 9999 9999 2 99995999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 '9999 9999 9999 9999 9999 9999 9999999c 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999

_____ 5999 9999 9999 9999 9999 9995 99 99_9999_9959 9999. _9999 999 99999 9999 9999

A6 4

ACCUMULATED At,'A YNESS :38 0 40 C 4 0 4 12 0 42 0

1 : n C 20 22 18 20 18 2 0 16 18 2626 14' 16 ' 12 " ' 14 1 2 '"" '14 6 12 .16 18 1416 14 16 6 14 20 22 14 20 1 8 20 16

, 16r, 220

240

20 22 0 0 0 0 n 0 0

CUBE ( 72 _j__82 _ IS s e l e c t e d .cube ( 1'.. , 106

_ .) HAS PEEN FOUND,

CHECKING pOINTS9 9 9 9 9999 999 9 0 9 9 9 9 9999 9999 9999 9999 9999 9999 99999999 9999 9 9 9 9 9999 9999 9999 9 999 9999 9999 9999 9999 99999 999 9999 9 9 9 9 9999 9999 ' 9 9 9 9 9 999 ‘"9 999" 9999 9999 9999 99999999 9999 9999 9 9 9 9 9999 0 9 9 9 9999 9999 9999 9999 9999 9 0 9 09999 9 99 9 9 9 0 0 9999 9999 9 0 9 9 9999 9 9 99 9999 9999 9999 9999

I 9 9 99 999 9" 0999! ACCUMULATED ANA YNESS •*

r 36 0 4? 0 4 0 4 12 0 42 n1 ' 0 (1 ?') 22 18 20 18 20 16 18 2628 14 16 12 14 12 14 6 1 ? 16 18 14!i 16 14 16 6 14 20 22 14 20 18 20 16

! / 18 22 2*4 2 o 22 0 0 ___ 0 n 0 0 0

CUBE ( 1 C"n

106' ’r.) IS selected.

CUBE < 14 , 10 )4

HAS BEEN FOUND ,cube ( 14 , 30 ) HAS BEEN FOu ND.wat ch for ... J - G . AND 30

! CHECKING POINTS,J 9 9 9 9 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999

9 9 9 9 9999 999° 99 9 9 9 9 9 9 99 99 9 W 9 “ 9 9 W “W 9 9 ~ 9 999~999 9 9'99 99 9 9 9 0 9 9 9 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999

".... '.9999 9999 9999 9999 9999 9999 9999 9999" 9999 9999.9999‘99999999 9999 9999

A65

A C C U !••!JL A T E D A a1 A V’N E S S :.J 3 6 0 4? 0 4 0 4 12 0 42 0

10 0 c 2 0 22 16 20 18 20 16 IB 282 8 14 16 12 14 12 14 6 1? 16 18 1416 14 16 6 14 20 22 14 20 ■10 20 16ibr .22.

024"0

~"2 0 " - 2 2 ~.o . .. o . 'O'" n.. .... 0 D_ "0

i 30 A! I S P E L E L ' T E D •

# •# F 0 R T R A * * S T O P / L O G C P TC E420 LOGOFF AT 1129 ON 11/14 , FOR TS|\ 4173,C E 421. TTpTTTfHriJSEEl CyoT6TT'86C3 SECTNUS'T

A 6 6

APPENDIX F EXAMPLE OF 8 VARIABLES, 126 MINTERMS

A6 7

TOATAL NUMBER OF ON-SET AND DONT«CARES = ON-SET = 126 DONT.CAr ES = 0NllMRFRS Op- VARIAJlll_r_____ B_________ ___

1 2 6

MMNTNT

0#

14

= 03

m ! NM in

,0.0

25

14

m IN.... MIN .#

36

e

M NT 0 7 6 MIN ,if 8 7 MIN .0 9M NT # 10 • 9 MIN ,if 11 = 10 MIN .0 12M NT 0 13 12 MIN .0 14 s. 13 MIN 15M NT if 16 - 15 MIN .0 17 - 16 MIN .0 18 -•M NT 0 19 18 MIN ,if 20 19 MIN • 0 21 22M NT 0 2? 21 MIN .0 23 _ 22 HINT.* 24M NT 0 25 24 MIN ,U 26 = 25 MIN .0 27 CM NT 0 28 - 27 MIN .0 29 - 28 MIN .0 30 22M NT 0 31 e 30 MIN *# 32 s 31 MIN .0 33 sM nt 0 34 £ 33 M IN 0 35 E 34 MIN ' 0 36 mm

H NT if 37 5 36 MIN 38 E 37 MIN :# 39 EM NT 0 4flL..E 39 . _ MI N .# 41 E 40 __MI.N ,0 42 sM NT 0 43 «• 42 MIN ,0 44 - 43 MIN .0 45M NT if 46 £ 45 MIN ,0 47 = 46 MIN .0 48 sM NT 0... 49 £ .... .48. .. ...MIN 50 S 49 j m i .1 0. 51 £M Nj 0 52 s 51 MIN .0 53 - 52 MIN .0 54M NT 0 55 e 54 MIN ,0 56 - 55 MIN ,0 57 sM NT a 58 57 MIN t# 59 s 58 MIN » 0 60 SM NT 0 61 £ 60 MIN .0 62 E 61 MIN .0 63 eM NT # 64 £ 63 MIN . it 65 s 64 MIN .0 66 £M NT 0 67 = 66 MIN .# 68 S 67 MIN .# 69 £M NT 70 f* 69 , MIN .0 71 E 70 MIN .0 72 £M NT f0 73 £ 72 , MIN ,0 74 E 74 MIN .0 75 £M NT # 76 £ 76 MIN’ .0 77 s 77 MIN .0 78 £M NT 0 79 = 79 MIN .0 60 E 80 MIN 81 £M NT 0 82 = 82 MIN .0 83 E 83 MIN .0 84 CM NT 0 8 5 85 MIN ,0 86 S 86 MIN .0 87 £M NT 0 88 S 88 • MIN ,0 89 s 89 MIN .0 90 £M NT if 91 £ 91 MIN .0 92 s 92 MIN .0 93 EM NT a 94 - 04 MTN .0 95 s 95.. mjnj:.0 96 £M NT 0 97 £ 97 MIN .0 98 s 98 MIN .0 99 £M NT 0 100 3 100 MIN .0 101 E 101 MIN .0 102 EM NT # 103 £ 104 MIN .0 104 E 105 MIN .0 105 5?M NT 0 106 3 107 MIN ,0 107 ■ s 108 MIN .0 108 £M NT 0 109 £ 110 MIN .0 110 = 111 min .0 111 £M NT if 1 1 9 11 3 MIN J 113 E 114 .... MIN j 0_114 EM NT 0 115 £ 116 MIN . if 116 E 117 MIN .0 117 CM NT 0 118 s 119 MIN ,0 119 S 120 MIN .0 120 £M NT 0 121 = 122 MIN ,0 122 E 123 MIN .0 123H NT 0 124 £ 125 MIN .0 125 = 126 MIN .0 126 £

2JL811141720..23.2629323538.414447505356596265_6J7175788184_879093

...9699

103106109112115lie121124127

A68

THE C U BE C O V E R I N G L I S T :

THE 1-TH CUBE COVERING IS I _____ 6-CUBE s ( 9 > 54 ) .= 0 0

THIS CUBE COVERS 0 1 2

THESE3

MINTERMS 1 4 5 6 7 8 9 10 11 12 13 14 15

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 3132 33 34 35 36 37 38 39 40 41 42 43 44 45 46 4748 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63

THIS MINTERM MT( 10 ) B 9 IS A CORE POINT

THE 2-TH CUBE COVERING IS I6-CUBE e ( 89 , 54 ) = 0--1----

THIS CUBE COVERS THESE MINTERMS I16 17 18 19 20 21 22 23 24 25 26 27 28 2g 30 31

______ 48 49 50 51 52 53 54 55__ 56 57 58 59 60__61 62_ 6360 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95

112 f 113 114 115 1 .6 117 118 119 120 121 122 123 124 125 126 127THIS MINTERM MT( 89 89 IS A CO~RE POINT

THE 3-TH CUBE COVERING IS «________________________________________5-CUBE = ( 77 * 62 ) =

THIS CUBE COVERS THESE MINTERMS I12 13 14 15 28 29 36 31 44 45 46 47 60 61 62 6376 77 78 79 92 93 94 95 108 109 110 111 124 125 126 127

THE 4.-TH CUBE COVERING IS I. 5-CUgE - ( 65 t 55 ) s O'— ~0 — 1

THIS CUBE COVERS THESE MINTERMS j1 3 5 7 17 19 21 23 33 35 37 39 49 5l 53 55

65 67 69 71 81 83 85 87 97 99 101 103 113 115 il7 119

A 6 9

THE 5-TH CUBE COVERING IS 1 5-CUrE = ( 72 , 50 ) = o— -0-0

THIS CUBE COVERS THESE MJNTERMS | 0 2 B 10 16 16 24 26 32 34 40 42 48 50 56 58

64 66 72 74 80 82 80 90 96 98 104 106 112 114 120 122

THE 6-th cube covering IS 15-qUBE b ( 75 * 62 ) = 0 — 1-1-

THIS CUBE COVERS THESE MINTERMS !10 11 14 15 26 27 30 74 75 78 79 90 91 94

3195

42106

43107

46110

47111

58122

59123

62126

63127

thei1

7-Th CUBE c^ E r INg IS f 5-CUBE s ( 70 # 29 ) = o-o- -1 —

THIS CUBE COVERS THESE MINTERMS 1 4 5 6 7 12 13 1^ 68 69 70 71 76 77 78 *5..79

2084

2185

2286

2387

2892

2993

3094

3195

i *

THEf

8-TH CUBE COVERING IS 1 5"tUrE b { 105 » 52 ) = 0-1-— o-

THIS CUBE COVERS THESE MINTERMS I 32 33 36 37 ‘40 4i 44 45 48 49 52 53 56 57 60 6196 97 100 101 104 105 108 109 112 113 116 117 120 121 124 125

THE FINAL CHECKING205 205 205 205 205 305 205 305 205 101 365 205305 305 405 405 305 305 305 305 305 405 305 405305 205 405 305 405 405 505 505 305 305 205_2 05.205 305 101 205 305 205 305 205 305 305 305 305405 405 305 305 305 405 205 305 405 305 405 305405 405 405 405 105 105 105 105__105 205__105 205105 205 105 205 205 305’ 305 205 205 205 205 205305 205 305 205 101 305 205 305 305 405 405 205_2.Q5____105 105 105__ 2 0.5_10.5___ 205__105 205__115_2 0 5___2 0 5205 205 305 305 205 205 205 305 101 205 305 205305 205 305 309 305 305

MAXIMUM CUBE DIMENSION =1 0 0 0 0 0 >♦) - - Am "I " ‘ “ 0 0 0 0 3 31 0 0 0 4 A 4 1 0 0 5 10 10 51 0 6 15 20 15 A 1 ____ __ __CHARAC TER Is u e s 44

6 5 6 b A 5 A 5 A 5 A 5 A b 6 5 6 5 6 1 6 56 5 6 5 A 5 A b A b A 5 A b A 5 6 5 6 b 6 56 5 6 5 A 5 A 5 A 5 A b A b 6 5 6 5 6 b 6 56 5 6 5 A 5 A b A 5 A 1 A 5 6 5 6 5 6 b 6 56 5 6 b A 5 A s “A 5 A b A 5 6 5 6 5 6 5 6 56 5 6 b A b A 5 A b A 5 A 5 6 5 6 5 6 5 6 16 5 6 '5 A nrO A b A 1 A b A :i 6 5 6 1 6 5 6 16 5 6 5 A b A b “A S'"'*A ..5 A__ b T~ 6 " 5 6 5 6 b 6 56 1 6 5 6 b A 5 A b A 5 A 5 6 5 6 5 6 1 6 56 1 A b A 1 A 5 A l A b A b 6 5 6 5 6 1 6 56 b 6 b A 5 A 5 A b A b A 6 5 6 5 6 5 6 56 b 6 b A 5 A 5 A 5

CUBE < 9 y b A ) HAS BEEN FOUND".

CHECKING POINTS9 9 9 9 9999 9999 9999 9999 9999 9 9 9 9 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9 9 9 9 9999 9999 9999 9999 99999999 9 9 9 9 9999 9999 9999 9999 9 9 9 9 9999 9999 9999 9999 99999999 9999 9999 9999 '9999“-9 9 9 -9 --9 -9 9 9 — 9 9 9 9 “9999 9999 9999 99999999 9999 9999 9999 9999 9999 9 9 9 9 9999 9999 9999 9999 999999f?9 9999 9999 9*999 16 16 16 16 16 16 16 16

16 16 16 16 16 ' ~.16" l u s r 16 16 16 16 1616 16 16 16 16 16 16 16 16 16 16 1616 16 16 16 16 16 16 16 16 16 16 1616 16 16 .. 16 16 :re~ 16“ 16“ T6 1 6 16 1616 16 16 16 16 16

ACCUMULATED AWAYNESS

CUBE

CUBE9

( 89

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 "0 “ D D “ ~ o 0 0 ... 0 0 “ 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 ~ u — 0 0 0— 0 "■ 0 D 0

0 0 0 0 Am 0 2 *3Am 0Am 2 nAii2 0A.. 2 '3 p n 2 2 n*— Am 0 0Am2 Am 9 fm) ~ n.-Am Am ')Am - 2 .--- 9 —Am --- ^ --Am .*3 ■' 92 0Am nAm 0Am 2 Am *3 OAm nAm 2 nAm2 OAm '■} Am 2 2 2 *3 OAm n

Am nAm 2 ~T} "A~ ..... r-j Am -.ry ■ ■" 'y '5 A ) IS SELECTED ♦

y 5 A ) HAS BELEN FOUND.

A71

CHECKING l-’OINTS999999999999999999999999

999999999999999999999999

999999999999999999999999

9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 32

9999 9999 9999 9999 9999 ' 24

99999999999999999999

32

99999999999999999999

32

9999 9999 9999 9999 9999 " ""32

99999999999999999999

32

99999999999999999999

24

999999999999999999993224

999932

999924

999932 24

9999 999932

999932

999999999999

99999999

99999999

99999999

99993232 32 32 32 "32 "32' 32 ' 32 32 32 32 3232 32 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999

ACCUMULATED AWAYNESSi *0 0 0 0 .. 0 0 ~ .“ T) • 0 3) 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 - 0 ....... 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 4 4 4 4 4 4 4 44 4 4 4 4 4 4 - 2 2 2 2 nAm

0 O 0A- '•> 0aL a— 9 n

/A. 2 2 2 ':> 4A A 4 A A 4 A 4 4 4 4 4A A

n n -C. Ai. 2 9A. 2 2 2 2 n

2 9 2 O OA.. '?Am

CUBE ( 89 y 54 ) IS SELECTE I.T. .....

ALL CORE CUBES HAVE BEEN SEARCHEDyAND THE MAXIMUM CUBE DIME:n s i o n i s ” : 6

CUBE ( 77 ? 2':> > HAS BEEN FOUND.CUBE ( 77 y 52 ) HAS BEEN FOUND.WATCH FOIR 22 AND 52 ...... ------------ — - — ■ ...

CUBE ( 77 y 55 ) HAS BEEN FOUND.WATCH FOR CUBE < 77WATCH FOR

62 ) HAS 'BEEN FOUNDT 62

CHECKING POINTS9 9 9 9 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999 9 9 9 9999 9999 9999 9999"'9999 "9999>"99999999 "9999 9999 99999 9 9 9 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999 9 9 9 9 9 9 9 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999

A72

f ; -V • \' \\>; i>9999 9999 9999 9999 9999 9999"9999 9999 “9999 9999 9999 9999/ 9999 9999 9999 9999 34 26 34 34 40 40 32 4032 40 32 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 "9999 "9999 9999 "9999 9999 9999 "34r 34 34 34 40 40 40 40 40 40 40 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999

s9999 9999 9999 9999 9999 "9999"

ACCUMULATED AWAYNES:3 :r 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0" 0 " '"O' 0 0 0 0' 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 _ _ ^ 0 0"' ‘ 8' 8 8 8 ~6 6 6 ~66 6 6 4 4 4 4 2 2 2 2 '3':> 2 oA. O 2 2 n n 2 88 8 8 6 6 6 6 6 6 6 4 44 4 2 2 2 '? n 2 9A. *3 22 o n 2 2 2

CUBE ( 77 ? 62 ) IS SELECT!;ID.CUBE ( 6 5 i’ 22 ) HAS BEEN F•o u n d :tCUBE ( 6 U s• 50 ) HAS BEEN Fr0UND<WATCH FOR 1 2_2I AND y50 ,CUBE ( 6 5 y 52 ) HAS BEEJTF■OUND, .......... - -

WATCH FOR SC» AND 52>

CUBE ( 65 f 55 ) HAS BEEN F'OUND.WATCH FOR 52! AND 55 --------------- --------------- — --------

CHECKING POINTSi

9999 9999 9999 9999 9999 9999 9999" 9999 9999 9999 9999 9999I 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999

9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999”~9999~~ 9 9 9 9 - 9999"- 9 9 9 9 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 42 9999 42 9999 48 9999 40 9999

34 42 40 9 9 9 9 - 9 9 9 9 - 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 429999 42 9999 48 9999 9999 42 48 42 48 9999 99999999 9999 9999 9 9 9 9 -9 9 9 9 - 9 9 9 9 - 9 9 9 9 - 9 9 9 9 - 9 9 9 9 -9999 "9999 99999999 9999 9999 9999 9999 9999

A73

ACCUMULATED AWAYNESS 5 "0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 “ 0 - 0 " 0 .0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 -- 0 ■ 0 o 0 ■•■~0 " "O' 0 0 00 0 0 0 1 0 8 1 0 8 8 6 8 6

1 0 1 0 8 4 4 4 4 o 2 2 'p o2 n 2 - ~ 2-- — 2

... ------------- ------ ~2 ■p *pAm 2 1 08 1 0 8 8 6 6 1 0 8 1 0 8 4 44 4 'p n 'PA- ^ o 2 2 'p 2

2 o 2 n 2 - n - ....CUBE ( 65 7 55 ) IS SELECTED♦CUBE ( 72 7 2 2 ) HAS BEEN FOUND.CUBE ( 72 7 50 ) HAS BEEN FOUND 7CUBE < 72 7 52 } HAS BEEN FOUND.CUBE ( 72 7 62 ) HAS BEEN FOUND.

POINTS 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999“*9999""9999“- 9 9 9 9 9999 9999 99999999 9999 9999 9999 9999 9 9 9 9 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 99W “ "9999 — ' 48 99999999 9999 48 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 56 9999 9999 9999 56 9999 56 9999 99999999 9999 9999 9*999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999

ACCUMULATED AWAYNESS J

CUBE ( 71

0 0 0 - “ 0 0 " 0 D ‘ 0" 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 - 0 0 " 0 " 0 — 0 — 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 10 8 10 8 10 6 10 6

10 10 10 " 4 4 4 4 . ~n -Am -- 2 — •pAm 2 2

'PAm 2 2 2 2 2 2 2 'pAm 2 2 108 10 8 1 0 6 6 10 1 0 10 10 4 44 4 nA— 2 . •p---— 2 — - - 2

*y.~ --- 2 ~ 2 •'PAm — 2

2 'p 2 2 2 2

7 50 > "IS SELECTED7---- ......

A74

, JV,| .... .-pi,!. W II I lp«^^u|l|IJ,»HiWlfl -II *w»wg*.v

CUBE ( 75 y 22

CUBE ( 75 y 50

WATCH FOR 22 AND

CUBE ( 75 y 55

WATCH FOR 50 AND

CUBE < 75 y 62

WATCH FOR h 'o AND

CHECKING POINTS

) HAS BEEN F O U N D ♦

) HAS BEEN FOUND,

50

) HAS BEEN FOUND,

HAS BEEN FOUND.

62

9999 9999 9999 9999

9999 9999 9999 9999 9999 9999 9999 99999 9 9 9 9 9 9 9

9999 9999

9999 9999 9999 99999 9 9 9

9999 9999 99999 9 9 99999

> 9 9 9 9 9 9 9 9 9 9 9

99999999999999999999999999999999

5899999999

9999999999999999999999999999999999999999

99999999999999999999999999999999999999999999

99999999"9999999999999999999999999999

99999999999999999999999999999999

64

99999999999999999999

589999999999999999

9999999999999999999999999999999999999999

99999999999999999999

569999999999999999

9999999999999999999999999999999999999999

ACCUMULATED AWAYNESS

CUBE

CUBE-

CUBE

WATCH

CUBE

CUBE

( 75

< 70

( 70

1 X ........- ■ --------------------------

0 0 0 1 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0.~ 0 — -0 — 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 . 0 0 0 0 0 0 0 0 00 0 0 0 “ 10 ”8 ... r o - ..." 8". 1 4 ”

5 _ T2 610 10 10 4 4 4 4 2 0 0Am 20Aii 2 9 n

Anp p 0

Amp 0An 2 9Am 10

8 to 8 14'- 6 . . .. . 6 " 10 “ “ 1 2 10 10 A 44 4 2 2 2 '? '•>

A. 2 n 0A~ 9AmnA~

'■) 2 nA. 2 2

y 62 ) IS SEE ECTE D.y 17 ) HAS BEEN F: OUND7 -------- ------------ ----------

) HAS BEEN FOUND.

FOR 17 AND

C 70 y 27 )

24

( 70HAS BEEN FOUND,

29 ) HAS BEEN FOUND.

A75

WATCH FOR 24 AND 29»W"jii.i«il!,Ji4L|.\uRJ!. pf fl'F'

CHECKING POINTS -9999 9999 9999 9999 9999 9999 9999 9999 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99999 9999 9999 9999 9999 9999 9999 9999 9999 9 9 9 9 9 9 9 9 9 9 9 99999 9999 9999 9999 9999 9999“9999~9999 9999 9 9 9 9 9 9 9 9 9 9 9 9

9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9 9 9 9

9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9 9 9 9

9999 9999 9999 9999 9999"9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 6 6 9999 "9999~9999 6 6 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999

ACCUMULATED AWAYNESS I

CUBECUBECUBECUBECUBE

( 70 ( 105 ( 105 < 105 ( 105

0 0 0 0 0 “ 0 ... 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 "0 "0" 0 ■ 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 10 8 10 8 14 6 12 60 10 10 4 - ij- 4 4 .-j 2 2 p r>A~

0 2 2 2 2 2 nA-

2 n nA’..

nA~ 10

8 10 S 16 6 6 10 16 10 10 4 44 4 2 2 2 .. 2— 2 2 0 2 2 2

n n nfU. tL. 2 2

9 ':>91

) IS SEE ECTE:d . ----

9 50 ) HAS BEEN FOUND..HAS BEEN FOUND ♦

55 )62 )

TI AS~BEE7'T "FOUNDTHAS BEEN FOUND.

CHECKING :'OINTS9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999 9 9 9 9 9 9 9 9 9 9 99999 9999 9999 9999 9999 9999 9999 9999 9999

99999999999999999 9 9 9 -999999999999999999999999

9999 9999 9999 9999 "9999 9999 9999“9999 9999 9999 9999 9999 9999 9999 9999 99999 9 9 9 -9 9 9 9 - 9 9 9 9 -9 9 9 9 -9999 9999 9999 9999 9999 9999 9999 9999 9999”9999~9999~9999“ 9999 9999 9999 9999 9999 9999 9999 9999 9999 "9999-----------

9999999999999999999999999999"999999999999

9999999999999999999999999999999999999999

9999999999999999999999999999999999999999

9999"999999999999999999999999999999999999

A76

' ACCUMULATED AWAYNESS I0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 10 8 10 8 14 6 12 6

10 10 10 4 4 A 4 2 na ! 2 n n

p o 2 '•> 2 p 2 ')A.. 2 2 10a 10 8 16 6 6 ~ 10 16 10 10 4 44 4 2 9 9 '•> nA,. 2 9 nA. ':>p '•> r'.> 9 9A* Am

- ... -..... -

f 52 ) IS SEL ECTED,>{<9##FORTRAN ** STOP

/LOGOFF B U T“A C E420 LOGOFF AT 1125 ON 11/09/77 r FOR TSN 0951, ' A C E421 CPU TIME USED J 000212, 3728 SECONDS *

All

APPENDIX GEXAMPLE OF 7 VARIABLES, 92 MINTERMS

A78

-Tr-rx

TOATAL n um be r OF ON-SET and d o n t «CARES = 92ON-SET - 92 DONT -CARES = 0NUMBERS OF VARIABLES = 7

MIN ,# 1 34 MINT.# 2 35 MINT.# 3 s 37MIN • it 4 = 39.. MINT.# 5 “ 44 —M-INI-a.# _ s ___ 47MIN , # 7 Z 48 mint ,# 8 — 49 MINT . # 9 s 50MIN , # 10 Z 51 MINT . # 11 — 52 MINT, it 12 s 53MIN .# 13 •j 54 m i n t .# 14 = 55 _M1NL.j# _ 15 - ___56MIN •# 16 z 57 m i n t .# 17 - 58 MINT.# 18 s 61MIN • a 19 Z 62 m i n t .# 20 s 63 MINT.# 21 s 64MIN ,# 22 = 83 m i n t .# 23 s 84 MI.NLJ 24 z _______ j 8 5 ___

MIN • # 25 = 86 MINT.# 26 s 87 MINT.# 27 z 88MIN . # 28 2 89 MINT.# 29 = 90 MINT.# 30 z 95MIN , # 31 z 1 MINT,# 32 - 2 MINT,# . 33_ _____3 .MIN . # 34 z 4 MINT.# 35 - 5 MINT.# 36 s 6MIN . # 37 Z 7 m i n t .# 38 s 8 MINT.# 39 z 9MIN . # 40 z 10 MINT . # 41 s 11 ...MINT..#..42 ._J!L___ .14MIN • it 43 = 15 m i n t ,# 44 r 16 MINT.# 45 17MIN , a 46 Z 18 m i n t .# 47 r 19 MINT.# 48 s 20MIN • it 49 - 21 m i n t .# 50 s 22 MINT . it 51 = ....23MIN • a 52 s 24 MINT, it 53 - 25 MINT.# 54 z 26MIN , # 55 2 27 MINT,# 56 - 28 MINT.# 57 z 29MIN . # 58 2 30 MINT.# 59 - 31 MI N T _, # 60 m.. 32

' " " MIN .# 61 S 96 MINT.# 62 - 97 MINT.# 63 - 98MIN <it 64 99 MINT , # 65 - 100 MINT. # 66 s 101MIN , i t 67 - 102 „ MINT,.# 68 r 103 MINT . # 69 - 104

... MIN 7?" 70 2 ’ 105^" MINT.# 71 s 106 MINT.# 72 a 10 7MIN f i t 73 Z 108 MINT , it 74 s 109 MINT.# 75 s 110MIN • it 76 s 1 1 1 mint.# 77 s 112 MINT.# 78 = 113MIN • it 79 2 114 m i n t ,# 80 - 115 MINT.# 01 s 116MIN • it fi2 - 117 MINT , # 83 - 118 MINT.# 84 - 119MIN • it 85 Z 120 . MINT . # 8 6 - 121 MINT.# 87 - 122MIN' • it 8 8 z 123 m i n t ,# 8 9 = 124 MINT.# 90 = 125MIN • it 9 1 z 126 m i n t .# 92 = 127 MINT.#

THE CUBE COVERING LIST :

THE 1-TH CUBE COVERING IS J________l^niRE = ( 44 # IQS L_=______-liUJLiU]------THIS CUBE COVERS THESE MH'TERMS i____4_4 IDjB— ________________ —

T H I S M I N T E R M MT( 5 > = 44 IS A C O R E P O I N T

th e 2 - T H CUBE COVERING IS s '1-CUBE = ( 64 i 96 ) = 1-00000

THIS 'CUBE_~C0VERS THESE MINTERMS I 64 96

T hT S m INTERM MT ( "21 64 fS! a" CORE p c I NT

JTHE 3-TH_CUBE COVERING IS } ____________3-CUBE = ( B3 * 55 ) = --10-11

THIS CUBE COVERS THESE NINTERMS :___________________83 55 51 87 19 23 115 119

J HISMINT E R.M_M TJ 2 2 L_?____ fl3______ I S. JLJLOR £... P Qi Nt..

THE 4-TH CUBE COVERING IS :4-C.U9E . =_____84 , 55 ) =_____ --1Q1--

THIS CUBE COVERS THESE MINTERMS I84 55 52 53 54 85 86 87 20

11^ 117 110 1*91

THIS MI NT ERM MT ( 23 ) r 84 IS A CORE PQIN T

THE 5-TH CUBE COVERING IS : 3 - CUBE =_(___ 89_»___ 56 ) = --HOP-___

THIS CUBE COVERS THESE M INTERMS J89 56 57 68 24 25 120 121________

THIS MINTERM MT( 28 ) = 89 IS A CORE POINT

THE 6-TH CUBE COVERING IS :3-CUBE = ( 90 # 56 ) s -110-0

THIS CUBE COVERS THESE MINTERMS j9o 56 58 88 24 26 120 122

T H I S M I N T E R M MT C 29 ) = 90 IS A C O R E P O I N T

A80

THE 7-TH CUBE COVERING Is t_____3-CUBE = ( 95 , 55 ) = --T-111---~

JHIS_CUBE COVERS THESE MINTERMS »________95 55 63 87 23 31 119 127

THIS MINTERM MT( 30 ) = 95 IS A CORE POINT

THE P-TH CUBE COVERING IS J __ 3-CUBE_ =_J____ 4 > 23 ) = Q 0 - 0 1 - - ____THIS CUBE COVERS THESE MINTERMS I ____ 4__23____5_____6 7 20___ 21___ 22________THIS MINTERM MT( 34 > = 4 IS A CORE POINT

THE " 9-th CUBE COVERING IS s3-CUBE = ( 6 * 27 ) = 00-10 —

THIS CUBE C Q~V ER S~~T HE SETM’lN'TERMST8 27 9 10 11 24 25 26

THIS M iNfERM MT ( 38 T ~ - 8 IS A CORE“'F6TnT

THE_ _ 10-TH CUBE COVERING IS :_____________ ;______________________4-CUBE = ( 14 , 19 ) = 00- — 1-

THIS CUBE COVERS THESE MINTERMS >__________________________________14 19 2 3 6 7 10 11 15 18 22 2326 27 30 31

T HIS MI NTERM- Mf< 42 14 IS A CORE POINT

THE 11-TH CUBE COVERING IS :_______________ ____________ _____ ___4-CUBE s { 28 » 19 ) = 001----

THIS CUBE COVERS THESE MINTERMS__ i_________________________________28 19 16 17 .18 20 21 22 23 24 25 ?627 29 30 31

T H I S M I N T E R M MT< 56 ) b 28 IS A C O R E P O I N T

* *A81

THE 12-TH CUBE COVERING IS :3-CUBE = ( 32 » 114 ) s -1-00-0

THIS CUBE COVERS THESE MINTERMS I32 114 34 48 5o 96 98 112

THIS MINTERM MT< 60 ) = 32 IS A CORE Pnj

THE 13-TH CUBE COVERING IS ; 5-CUBE = ( 97 , 126 )._a______11

THIS CUBE COVERS THESE MINTERMS t_? 7 126 96 98 . 99. _10fl 101 102 1Q3 104 1 ns i n 6

107 108 109 110 111 112 113 114 H 5 H 6 117 U 8119 120 121 122 123 124 125 127

THIS MINTERM MT < 62 ) = 97 IS A CORE POINT

THE 14-TH CUBE COVERING IS :__ ______________________ ____4-CUBE = ( 49 * 118 ) = - H O —

THIS CUBE COVERS THESE MINTERMS I___________________ _________ _____49 t118 48 50, 5l 52 53 54 55 ll2 113 U 4

115 116 117 119

THE 15-TH CUBE COVERING IS t3-CUBE = ( 47 . 119 ) = -1 — 111

THISCUB e COVERS THESE MINTERMS *47 119 39 55 63 1C 3 111 127

the 16-th C u be c o v e r i n g is s3-CUBE = ( 37 I 119 ) = -1-01-1

THISCUB e “COVERS THESE MINTERMS »37 119 39 53 55 101 103 117

A82

THE 17-TH CUBE COVERING IS '*......... .3-CUBE = ( 1 r 23 ) = 0O-O--1

THIS CUBE COVERS THESE MINTERMS j1 23 3 5 7 17 19 21

THE 18-TH CUBE COVERING IS J3-CUBE = ( 35 , 119 ' ) = -l-O-li

THIS CUBE COVERS THESE MINTERMS I35 119 39 51 55 99 103 11.5

THE 19-TH CUBE COVERING IS J3-CUBE = ( 61 , 119 ) = -11-1-1 1

THIS CUBE COVERS’THESE MINTERMS I61 119 53 55 63 117 125 127

THE 20-TH CUBE COVERING IS :3JCUBE = ( 62 f* 119 ) = -11-11-

THIS CUBE COVERS THESE MINTERMS I62 119 54 55 63 llfi 126 127

THE FINAL CHECKING105 105 105 305 101 105 205 105 205 305 205 405

_________ 305 9_05__205__105 105 105 105__4p5___!Q_1__2J1__101 1 0 1 ____;101 305 205 101 101 101 105 105 205 101 20l 201 i305 101 101 201 201 101 101 105 205 205 405 305 I405 _ 4Q5 705 405 305 4Q5 301 101___10.1__205__3.0.5__ 101___!305 101 205 205 101 2 05 101 405 101 101 101 101205 101 101 205 305 205 305 405 305 505 4o5 1005 '305 205 205 101 1 0 1 2 0 5 2 0 5 505 _____

A83

V v V V U UCOCO r\ /n * *7 7 7 7 . i « £ * f M

J .

0v »♦ o0

0 © 2** 4 0a _v/ 0

A AX V AX *t a AX V •* Ax n 0 A Ax n A /AX V A Ax n 14 18 A AtXV «* AX n

14 •1 A»XO A AXV J Axn A Axn 14 14 1 o o lo 10 A AX na *X *t lo J AX V A Ax n A Ax*t A / AXO 12 12 1 o 8 12 12a AX o 8 12 12 1 o 12 1 o A /AX/d 16 lo 20 12

-•J /xo •i /xo /A A/ClV 12 j. o <A /XO /"A A/dV •J /AX /d 1 o lo 20 o

A*t Ato 8 A /AX aCI

An / Ao 8 12 4 8 8 12- - - A

V. — ___jT\

o__ n .O — __ M / AX Ail

--------------- n . -O _______d f=k----------X '12 ~ ~ - 1 o J 8 A A tX/d A A tX /Cl A AXOr%o 4 A\x .d A <AX-d a /X o /Ao A /AX/d A / Ax*d 1 o

CUBE ( / Aon A> Ay 7 0 A

f 13 / A l - I I - / AOLLC.U » r t ~ r .i rr.JL> ♦

CUBE ( S3 y 33 ) I I A Atnno BEEN ECU ND t

C H E C K I N G f \ I Mrruxiy io2 An AtV /Ao A t r \ / A A\7 7 7 7 1 4 o /Ao 7 7 7 7 3 AtoS 7 7 7 7 4 .

AX X 3 3 8 A t A t A t At7 7 7 7 A t A t A t A t7 7 7 7 2 4A*t 7 7 7 7 («X 0 0 2 /C. 4 Ato 0 At

4 .A t

s 0 0 AV /•% AX 2 4 Ato A to A t At A t At7 7 7 7 2

o O 7777 AV 8 1 At A t 8A t A t / n7 7 7 7 An 4 A AX V n 6 o 8 An An 47 7 7 7 n-' / A An 4 A /AXV A AX V 7 7 7 7 o 8 8 A t A t A t At7 7 7 7f 4 /*>

/u./ A V

18 / A A 4 / Ao

HULUriULM 1 C-JLI HWH 1 1X1“ oo ♦

a aX *t •i /X o A Axn 16 AtV 18 A AX *t A AX o 1 o 18 14 1 o16 •i nX o a tXO A /AX o 18 lo 18 / A A/dV o 18 A Axn 1 oa aXO xo A Axo ' A /AX o 18 /A A/dV lo «• / XO 18 A Axn 1 o 1 oj / nXO 1 o A / Axo A /AX o / A /A/dV 18 / A A*d V A AX o 18 A A tXO 20 A AX oj / nX o U /AX o A t /A/dV a a \X o / A A/dV / A /A

/ d V/ A /A 18 A t At/Cl V A t At

/ C l Vc c 12

An x 2 J /AX/d A Ax n •i AX V A /AX/d •4 /AX/d A Ax n A A tX /d a Axn 14 160 J Ax*t A AX *t A AX o 12 A AX *t a ax n A AX o •1 A t•I* Am

A Axn 14 A AX o•I AX n J Axo A AXO 1 o 14 lo 1 o 18

r%» * r . r " /LUDC. k r \oo y 33 ) *r / aX o 3 ELEC T E D ♦

m i r . r " aLUDC S. n aon y 33 ) • 1 A / AriMO BEEN F O U N D ♦

C H E C K I N G PGINTo3 3 A AX o / A / A A t / A

7 7 7 7 1 12 A AX o 16 A t A t At A t7 7 7 7

A t A t A t At7 7 7 7

A t A t A t At7 7 7 7

7 7 7 7 7 7 7 7 3 / AAm

—A/

—A/ A AX o A t A t A t A t7 7 7 7

A t A t A t A t7 7 7 7

A t A t A t A t7 7 7 7 7 7 7 7

7 7 7 7

X O/ n r » / n / n7 7 7 7

n r \ r > / A /a7 7 7 7 -cl

/A /AV V/ A / A / A / A ^ < A / A / A7 7 7 7 7 7 7 7

AV/AVaC»

AtVAM

A*t/ AAm

A

" toAn

A /AXV

o 12 3

A AtX VA AXO10

4 . A AX o 10

A AtXVA t A t A t At7 7 7 7

A Axo

10 / t A t A t A t7 7 7 7

A t/d

r-.'V Jf; •

A84

W A A 4 *rI’l M A X nun CUBE DIME No!

4X

Au 0 0

AX 0 rx

u4AU

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4o x 3

THE CH m R m CTERI o FICo3 X 3 1 A

M4

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/ a i t v . r - v . t % /1~* v i nuujdc. juxnc.no,

3 310 3

X O . L X X * t X O » J O X O U o l o l

^ \ j X \ J X v J X \ j X v j x o x l l 3 ij t— t— A .1 "V .1 "Y J "T J A .1 r— ^

X x J v J M X O X O X O X n X J X O ^ J4 / 4 "▼ X * •“ J r “ 4 * 4 r - 4 w JX O X O X M X U X U X M X U X u l

\ J J vJ O vJ O J O J J J vJ J O J J XJ / |~ "V J l~ »“ I - 4 4 4 / 4 r - 4 •> fX O s J O X v J \ J v J X O X O X % J X \ J v J

X \ J X vJ X vJ X \ J vJ ^ J X sJ X vJ \ J O X5 6 3 6 3 6 3 6 3 6 3 6 3 6 3 3 14 4X o

L U D L

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V V VA A AV V V0 0 0

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1 A A l i v n II a • r i - v . a i i a a%/a *MUUUriUL-M I C.JU MWH I IMC.OO ♦4 *O O *f6 8 4/A 4 A 4O XV o/A A /O *f o/A /A J AO O X VA 4 *m o oA /A /Au ^ -c:A * *M O O

O M r 70

urmunxnu ruxivioAX 0 0A A /AU u Am

A A 4U u XA A S\u u U

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1 3 1 3 1 3 1 3 4 4X o 1

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r—vJ

BEEN LOWERED TOJ 3

H m S r . r - i - s iJDC.C.IV F O U N D ♦

0 9999 4X

Au 0 0 0

0 0 /A 0 0 00 0 A

V 0 0 0 04%u 0 1 0 0 0 4\

uAu 0 4%

uAV 1 0 4%

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Au

/-A 4*% 4A 4*%7 7 7 7 1 0

Au 0AU 0 /A

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A85

/nr>/n/n7 7 7 7 / A

o o 1 2 x 2 1 4 1 4 l o 4 4 4 27 7 7 7 A An n

/o 1 2 1 8 1 o 9 9 9 9 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 ?a

OA An n 1 0

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■V I— v*. AIIASASI r*~ r\ An A1 C.JU M W H 1 I V I X O O ♦

A MSJ. o 2 0 1 6

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a msX o 1 8 1 o

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o 1 3 1 4 l oA AX o

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7 7 7 7/ n / n / n n7 7 7 7

/ n A n A n A n7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7

7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7

7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7

9999 7 7 7 7An m a i An7 7 7 7

A n A n A n A n7 7 7 7 9999 A n A n A n A n

7 7 7 7 9999 A n A n A n An7 7 7 7

A n A n A n An7 7 7 7

A n A n A n A n7 7 7 7 9999 9999

7 7 7 7r \ r \ / n / n7 7 7 7 9999 A n A n / n A n

7 7 7 7A n A n A n # n7 7 7 7

A n A n A n A n7 7 7 7 9999 9999

A9 4

HLbUriULH I C.JL* HWH I HC.OO ♦

UUDC*7/ r

A f'XM O 6 2 6 0

1“ /AJ O

AV 6 2 4 0

j /X o 1 8

/A /AA^V 2 2

/A Aa£M 6 4 6 6

a *x o l o

/"A /Aa£ V

AA fA *!_ All. 2 4 3 0 6 6

3 o 3 8A A

M VA A

M VA /*\

M £ 4 0A

M a£rf /AX O l o 2 0

/A /A Am Am

/A *-clM 2 6 3 6

AM

r*~ /aJ O 0 4

1" /*\ J O 3 6

* *> o v 3 6

0 6 2t—* /-AJ O

/ /AO a£

A *M O

r— /aJ V

A tM O

1 o 2 0/A /*A 1" /A

vJ a£i— f\J V 34 1” A

J V

/ /nO a£ j» •1 A /AX X 7

\/ •r /*\X o 3ELEC 7 EL ♦

o i ur

M O30« AMV3434t A0 V1 o/ n r\zo

MV

o42363636

x o1 o34 34 34 o 0•» AX M 16

/ i / a / a n •*" r ~ r . i i t/ Luuurr d u ir \ r—’ r - * *r a* a /a *HI a V^M

CPU TIriE USED*r*.t r~ a n r * i n / A r \ MrL.ir.MOLi Luuuiy*

UIN r u r \ i c m

f1

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A95

APPENDIX H

EXAMPLE OF 9 VARIABLES, 120 ON-SET MINTERMS, AND 40 DON'T CARE MINTERMS.

TOATAL NUMBER OF ON-SET AND nCN'T*CARES = ON.SET = j.2o DONT^CARFS = 4 0-NilMBgRS-uF-.VAR I ASLf S = ID-------------

M NT 7 s 7 MINT # 8 = 9 MINT • a 9 =M NT # 10 = 12 MI N'T if li = 13 MI N'T if 1? s

--------------------M NX fl­— 13— *__ -16- _. wlNX J t__1-4— s_-17— .-.m i n t , f l 15- .. sM NT it 16 = 19 MINT # 17 = 21 m i n t . i t 18 sM NT # 19 = 23 MINT if 20 - 24 MINT . if 21 -M N'T 29 • ?7 M J M T JJ ? 3 2.8. __MI.NX• H -24- _

M NT Ifit 25 - 30 Ml N'T •"If

if 26 31 m i n t a 27 =

M NT # 28 = 35 MINT if 29 - 36 m i n t .f l 30 =

M NT # 31 - 38 MINT if 32— - - 3-9__ - m i n t # -33 ■»

M NT # 34 42 MINT it 35 43 m i n t .# 36 -

M N’T # 37 s 48 MINT if 38 = 49 ■ m i n t • it 39 sM mT 40 • M T N T if 41 52 ..M I NT it 43. -

M NT # 43 = 55 MINT if 44 56 MlNf . # 45 —

M NT # 46 z 58 MINT it 47 = 59 MINT , t t 48 sy N'T ii 49 61 M I NT tt 50 _ 62 MINT-■*4 — Si- _M NT it ~

52 2 64 MINT it 53 z 68 M I NT •# 54 sM NT # 55 r 70 m i n t a 56 = 71 m i n t # 57 r

............M NT # .5.8 s 7.5 .... ____ Ml NX. # .....5 .9_ 76 . MI N.f-;# ...AO. s

M N . # 61 = 78 MIN . # 62 79 m i n t # 63 =M N • a 64 = 81 MIN .if 65 = 90 MINT A 66 =M N'_ *# -67-.=.. ...92.. ___ JUN • u . 68 = 93 _ ..MINX--•# .. 69 sM N if 70 = 95 mi n it 71 96 m i n t # 72 =M N . i t 73 = 98 MIN . i t 74 -s 103 MINf # 75 -

K N .it .7.6 = ..__ U jS_ ... MIN: .f l 77 s __.!M ..JUilT- it 7fi z

1X

MINJ UX

2_5_

160

2_5_

A96

MINH IM

3A.

1114IB2226-2-93437414550-5357 60 63 69 72 23. 80 91 94 97

104 107 111 114 117 123 126 129 152 157 160 163 167 .172 175 179 167 190 201 _2X4_

M NT it 79 = 108 MINT if 80 = 109 MINT it 81 -M NT it 82 = 112 MINT it 83 r 113 mint it 84 sM NT '# .85 S 1.15 -.m i n t .JL— 8-6-_=_... 116. __MXN-T. H 87 .=M NT fl­ 88 s 121 MINT tt 89 s 122 M I NT it 90 =M NT it 91 = 124 MINT if 92 = 145 mint it 93 cM NT # 94 127 MINT it 95 128. _ MINT ti­— 9-6— _zM NT it 97 s 150 MINT it 98 151 MINT lt 99 -

M NT U 100 s 153 m i n t a 101 - 156 Ml N'T it 102 sM NT it 10J5 2 J.58 MINT it 104 z 159 .HINT fl­ .105— =M NT it 106 = 161 MINT it 107 z 162 MINT it 108 z

M NT fl 109 C 164 MINT it 110 - z 165 MINT it 111 zM NT n 11? = 168 MINT it 113 . m 170 MINT it 114... .z.

M NT it 115 s 173 MI N'T it 116 Z 174 MINT it 117 z

M NT U 118 s 176 MINT it 119 = 178 mint it 120 -

M NT it 121 = 185 MINT U 122 z 186 MINT- fl-123— z

M NT it 124 JJ 168 MINT it 125 z 189 MINT it 126 =M NT it 127 5 191 MINT it 128 •;S 192 mint it 129 SM NT it -13-Q..... s .. 20.2 . .........M_I.Nl. it 131 S 20.3.... ...MINX. it 13.2 .N N . i t 133 5 205 MIN ,f l 134 S 206 MINT.# 135M N . it 136 s 213 MIN . i t 137 = 214 MINT.# 138 =M N . fl 139 a. 216 MIN .fl 140 217 — MINX# 141 -M N • it 142 = 223 MIN »# 143 s 224 MINT.# 144 sM N .f l 145 s 226 MIN .A 146 . c 227 MINT,# 147 =M N , 4 1 48 - 229 MIN .f l 149 s 230 MI Mi. JL 1 *snM N . i t 151 246 MIN ,fl 152 s 247 m i n t .# 153 =M N • it 154 S 249 MIN . i t 155 s 256 m i n t .# 156M N . i t 4.57 258 MIN ,f l 1 58 s ._259 MINt ,#.. 157M N . i t 160 = 260 min ,f l

207215220225

246257254

THE CUBE COVERING u IST :

ThE 1-Th CUBE COVERING IS I— -.....3-CUBE .= < .81 , 48 ) S OOO--IOQO-___

THIS CUBe CqVeRS THpSe MlNTERMS t-81 — 1-8— 1-7— 4.9 8-0— 1-12— 1-1-3-----------------------THIS MINTERM MT( 64 ) = 01 IS A CORE POINT

THE 2-tH CuBE COVERING IS J4-CUBE = ( 90 » 63 ) = 000--U-1-

THIs C^BE COVERS Th Ej E M i^TEr Ms •90 63 26 27 30 31 58 59 62 91 94 95 122 123 126 1?7

THIS MINTERM MT( 65 ) = 90 IS A CORE POINT

I r-T+IE......3 - t-H -Gy B F— CO-V^RIN G— IB— «................................ .........

4-CUBE = ( 156 , 63 ) - 00-0-111 —

-THIS CUBE COVERS THESE MI-NIE.RmS-.-J - - _____156 63 28 29 30 31 60 61 62 157 158 159 188 189 190 191«

-THE 4 -JH._CUB.E-. COVERI ___________________3-CUgE = ( 38 , 5 ) s 0000-001--

THIS—C4J-B-E-.C 0 V£RS - T HESE— MIN T-E RM-S-.. I________________38 5 4 6 7 36 37 39

IRE_____ ft-TH CURE COVFRTNft T R I__________________3-CUBE = { 162 , 51 > = 00-01-001-

-T .-H IS -C U B E .C O V E R S T R E S E - M I N T E R M .S 1 __ _ _ _ _ _ _ _ _ _ _ _ _162 51 34 35 50 163 178 179

A9 8

THE 6-TH CUBE COVERING. IS I ........3-CU0E = ( 41 , 123 ) = OOO-l-lO-l

IBIS-C U B E C QA/E5S—JHE-SF... M I N T F R MS_J________________41 123 43 57 59 IO5 IO7 121

THE 7-TH CUBE COVERING IS I3-cUBE = ( 69 , 78 ) = 000100-1 —

IHIS-CUBF C-QV-ERS-T-HE-SE.. M IKTFRMS S________________69 78 68 70 71 76 77 79

-THE - - 8-TH CUBE COVERING IS I . _3-CUBE s ( 164 , 225 ) = 001-100-0-

- T H I S - - C U B g -C Q JV -E H S—T H E S E — M 4 - N T E _ a j iS _ j _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _164 225 160 161 165 224 228 229

THE 9 - 7 H-CUB S - C O VERIN-G— IS J-----------------—3ICUBE = ( 150 ;# 31 ) = 00-001-11-

THIS .CUBE -COVERS-THESE JlUlXEJ?« S _ | ________________I5O 31 22 23 30 I5I 155 1^9

THE_____ 1.0 - T-H -CUB E-CO V ERINC IS I_____________ __3-CUBE s { 174 . 189 ) = 00101-11

TH-I-S —C UBg— C Ql/S RS—THESE— M I N TERMS 1______________174 189 172 173 175 188 190 191

T H E _ _ _ _ _ _ l l - T H — C l l B E - C O V E R I H Q - X S — I_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _3-CUgE s ( 116 , 49 ) s 000-110-0-

T H I S C U B E C O V E R S T H E S E . . M L N T E R M S i , ___ _ _ _ _ _116 49 48 52 53 112 113 117

A99

THE lz-TH CUBE COVER I MG IS i ..........3-CUBE = C 2 , 51 > = OOOO--OOI-

mJ-S ■ ■ C UB F-—C-O-V-gfi-S— T-HESg—-MXNT EB MS_J---------------2 51 3 18 19 34 35 5O

THE 13-TH CUBE COVERING IS I__________________3-CUgE = ( 170 » 50 ) = 00-01 — 010

THIS CUBE COVERS THESE MINTERMS I_... ....... ...170 50 34 42 58 162 178 186

THE 14 -TH CU3F COVERING IS ?___________________3-CUgE = ( 12 # 93 ) = 000-0-110-

THIS CUBE COVERS THESE MINTERMS :12 93 13 28 29 76 77 92

THE 1 9-TH -CUBE- C0V-ER-I-KK?—-1-5— I-------------------3~cUBE r ( 75 J 59 ) = 000 — 1011

THIS CUBE COVERS THESE MINTERMS I........... ....I 75 59 11 27 43 91 10 7 123it_________________________________________________________

-THE 16-TH CUBE COVERING IS I . ............ ...3-CUgE = ( 98 # 58 ) = 000-1 — 010

—-T H -I-S—CU B E — C Q -V E ^ S —T H E S E — MT-NT-E-R-M.S— i - - - - - - - - - - - - - - - - - - - - - - - - - - - -98 58 34 42 50 106 114 122

T-HE-- - - - - - 1-7 - - T-H —C U B E — COA^E R I N G I S — I- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -3-cUbE = ( 45 # 21 > = 0000 — 101

T H I S - C U B E - - C O V E R S —T H E S E —M 4 -N T E R M S - j - - - - - - _45 21 5 13 29 37 53 6l

A 1 0 0

'l f

t he 18-th c u b e c o v e r i n g is »3-CUfiE c ( 14 * 68 ) = 000-00-1-0

T -H -l-S—C U B_ _ _ _ -0-V -& R S—T - H £ S £ — M T U T E R f l S _ t _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _14 68 4 6 12 70 76 78

T-HE_____ l^Jj^UCueE-C-OV-EJS-UxlG— LB— t-_________________3-CUBE = ( 24 , 6 2 ) = 0 0 0 0 - 1 1 — 0

T-H IS-C-UB E-C-CV-EES-THE S-E-Hl-NJ-EfiH S_. j ___________24 62 26 28 3 0 56 58 60

THE ___ 20-T.H -CUB.E .. CAVER ING— IS .. I________________ -3-CUBE * ( 1 * 23 ) = 0000Q-0--1

-T-H4-S C Uflg -CqvmS-T-HESE M I NTERM-5._i________________1 23 3 5 7 I 7 I9 21

T H E 2Jl^ T H — CUBE COVERING 1-S ___________________3-CTUgE = ( 111 , '* 91 ) = 0001 — 1-11

THIS CUBE COVERS THESE MI N’T ERRS \111 91 75 79 95 IO7 123 127

~rnL _ J

— --- THE 2grTH CUBE COVERING IS I___________________3-cUBE = ( 124 # 30 ) = 000--H1-0

THTS-CU BE COVERS T HES-E— Kl-N T-E-R-M £--»----------------124 30 28 60 62 92 94 126

THE 2^-TH-CUBE-COI-VE-R I NG IS -»-------------------3-cUBE = ( 115 , 48 ) = 000-1100 —

THIS.-CUBE-COV ERS-TH E S E-M1NXE.RM-S-.1 ---------------115 48 49 50 51 112 113 114

A101

-THE ------24-TH -CUBE COVe RU'G . IS !_________3-CUgE = ( 55 # 17 ) = 0 0 0 0 - 1 0 — 1

j 3 m J - S _ C U B m n . U F R S t h f r f m t n t f r m -S t_ _ _ _ _ _ _ _ _ _ _ _5 5 I 7 I 9 21 23 4 9 5 i 5 3

T-HE______2 5 =XH— C-U B E—CUV ERXN G— l-S_I__________________2 -cUBE s ( 128 # 161 > = Q0 1 0 - 0 0 0 0 -

IHi-S .CUBE .COVERS. ..THESE ...MI N T E R M S 1___ _____ _______128 161 129 160

,-JLH.E 26-TH CUBE COVERING IS_i________2-cUBE b < 103 # 7 ) s 000 — 00111

-T H IS CUBE COVERS THESE MINTERMS »________________103 7 39 7l

THE______ 2 1 - T H - ..C UJ3-E _C.Q.V£Bj MG _I,5—«_____________________2- ' c UBE s ( 72 9 6 ) = 0 0 0 1 - 0 ^ 0 0 0

THIS CUBF CnVEPS THESE MINTERMS \72 96 64 104

THE _ 28-TH CUBE COVER IH_G_J S \__________ _______2-CUBE b £ 168 * 162 ) = OOlOlO-O-O

THIS CUBE COVERS THESE MINTERMS 8________________168 162 160 170

THE 29-TH CUBE COVERING IS (2-CUBE = ( 9 , 5 ) = 000O00--01

T-H-I-S CUBE COVERS THESE MINTFRMS-i----------------9 5 1 13

A 1 0 2

THE -SOrTH c u b e c o v e r i n g IS I 2-CUgE a ( 145 . 1 > =

W « - C 4 ^ € - W W S - ^ E S e - 4 i W f E 5 M S _ 4 145 l 1 7 1 2 9

THE---- 3i._tH-XTU0MCVERIHG-I-S - i _2-CUgE = < 152 » 217 ) =

TRlS-CBR E_CT1V_ERS ._lH£S£_IHNlEaM5_J 152 217 153 216

THE---- 3 2 = T W.CUB E -CO VERI-MG -I S._ 1 ___________ _- - .2-cU0E a ( 108 . 105 ) = 0O0HO1-0-

-TH-I-S -GU B E— G-QV E-R-S—TH EBE-MTHTERH-S—I----------------108 105 104 109

-THE----- 3 3 .-TH _CH0E-COV-ESfULG-XS-T----:-------------2-cUBE a ( 176 / 50 ) = 00-01100-0

THIS -CUBE COVERS THESE MJNIERm S I — ____ __ -176 50 48 178

THE_____ 34HTH--C-UBE— COV-E-S-I-M-G-. I5-J__________________2-CUb E a ( 167 # 37 ) = 00-01001-1

-THIS CUB-g— CG-VE-HS—THESE— M INTER MS I----------------167 37 39 165

THE 35t TH CURE--.CCVE-RIMG IS »___________________2-cU0E = ( 97 # 224 > = 00-110000-

-THIS CUBE COVERS THESE HIHIERKS ..1 _ ....................

97 224 96 225

00-00-0001

001-01100-

A103

-r-ME -F I -NAL—C44E-OX4-MG ________________________________ — ___________________________________305 105 205 205 405 205 305 105 105 205 305 1 0 5 ™105 405 105 305 305 105 305 105 205 205 405 305

_________ 5.05. _ 305 405 _2Q5_ _ 105__ 305__ 105__ 305__105_ 205__205__JC5405 405 605 '405 105 305 105 105 105 ~405 305 305205 405 205 105 205 105 205 205 105 205 305 205205 205 105 101 101 301 205 105 205 205 205 105105 105 205 205 105 305 105 105 105 305 305 205105 105 105 105 205 405 105 105 205 201 105 205

____________ 105-- 105-- 105__ 105__ 105__ 1115__ 2J15__ 245__ 345__ 205__3 05__105105 205 105 105 205 105 105 105 105 105 305 105109 209 109 309 205 205 205 109 109 109 109 109

____________ m s — m s — m s — m e m s m s 20s— 24s— m s — m s 2^5 sss109 109 209 209 109 109 109 109 109 109 109 109109 109 109 109

A104

m a x i m u m c u b e

.. - 1 .0.DIMENSION_Q_. Q . n

70 n 9 1. n n

3 3 1 0 0 0 0 4 6 4 1 05 10 10 5 l 0 0 15 20 15 6------ 7--- 2 4--- 35- — 35--- 2-1— ■J— — 1

01

0p0

T H E C H A R A C T E R I S T I C S j

r ... ..... 6 ... ..5...-1 ...6 .5 ... 6 . .1.. 6 .5. 6 5 6 . ..4.. .5. .1 . 5 1 64 5 1 6 5 6 1 6 1 5 1 5 6 5 1 6 1 A6 6 5 6 5 6 5 6 5 6 1 5 6 4 1 6 5>5 A 1 A 1 A 5 6 1 A * A <5 A e; 5 K6 6 5 6 5 6 5 6 5 6 1 7 5 5 1 4

—A — 1 4

5 3 1 5 1 6 5 6 5 6 1 6 4 3 1 4 1 56 — 4 __1.. .. 6 _6 .5 6 _1.. .4..._1_. 5 3 . 5 1 6 1 86 4 1 5 1 5 1 6 1 5 5 5 4 4 1 4 1 56 6 5 5 1 3 1 6 1 5 1 3 4 4 1 4 1 4*5 * e; * «s i A 1 «S 5 5 A e; «5 A ts,4 3 1 5 1 5 1 5 1 4 1 4 3 5 1 4 1 55 5 1 5 1 6 1 6 6 1 3 3 2 1 4 1 4

. 4 . ... 4 1 5 1 ? 1 3 1 4 5 4 4 4 1 4 1 A5 5 1 3 1 4 1 4 1 3 1 4 4 3 1 2 1 43 3 1 3 1 3 1 3 1 2 1

M A X I MU M c u b e d i m e n s i o n h a s b e e n l o w e r e d t o S

_M A X^UJl-^UfiE-D-l-MEN Sl^NT HA_s _B£EN- -LOWER ED— IC J_

CUBE ( 61 , 48 >' HAS BEEN FOUND,

6- 5 -

C H E C K I N G P O I N T S8 1 2 2 2 0 0 0 0 0 0 n

. .................... 9 9 9 9 . 9 9 9 9 4 8 s 4 1 2 4 p _____ 2 . .... 2 . .20 0 2 2 1 2 0 0 1 0 0 n

9 9 9 9 9 9 9 9 6 8 4 8 1 4 8 2 2 2? fl 0 4 ■1 n 0 n p n (1 ..........J1

0 0 9 9 9 9 9 9 9 9 1 2 2 1 0 0 8 42 0 2 2 0 0 0 0 0 9 9 9 9 9 9 9 9 R4 4 4 8 ? 2 2 2 n n 0 ... 10 0 1 2 0 0 0 0 2 2 0 n0 0 0 0 0 0 0 0 0 2 2 i

9 0 9 0 0 9 0 9 0 0 9 0 0 9 0 9 9 0 0 0 0 0 0 9 9 0 9 9 0 9 0 9 0 0 9 0 0 0 0 0 9 0 0 0 . 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 0 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 0 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 0 9 9 9 9 9 9 9 99 9 9 9 0 9 9 9 9 0 9 9 0 9 9 0

a c c u m u l a t e d a w a y n e s s•• -

2 4 4 4 4 6 6 4 6 6 6 P0 0 2 2 2 4 4 2 4 4 4 4

6 _____ 6 4 4 .. . ,_ 4 _ 4 6 ........ 6 . .... 4 . ____ 6 6 A

A105

0 0 2 2 2 2 4 2 2 4 4 44 6 6 2 4 4 6 6 4 6 6 6.......... -.. 8 R 0 n 4 4 4 A, 4 p4 6 4 4 6 6 6 6 8 0 0 -

?2 2 2 2 4 4 4 2 6 6 4 46 . 6 .4._— J. .... 6 __ 6__ 8 . 8 .. 4. 4 6 66 6 8 6 8 8 8 10 10 2 4 40 0 0 0 0 0 0 0 0 ft 0 n

.............. n n .. Q .... fl n 0 0 0 ft ft ft n0 0 0 Cl 0 0 0 0 0 ft 0 n0 0 0 n

CUBE ( 81 , 40 > IS SELECTED,CUBE ( 9'j , 63 ) HAS BEEN FOUND,CHECKING POINTS

----------------8---- 2-----4-----2--- -2-----2-----2-----Q-----4-----1---- 2---- 2-9999 9999 16 16 5 5 10 8 9999 9999 10 109999 9999 4 4 1 2 0 2 2 4 8 ?

------------ 99 99-9999-1 6 —— 16- -5------1-0----- 5---- 8---16— 9999 9999. in10 9999 9999 4 1 0 0 1 0 4 2 ?8 8 9999 9999 9999 9999 10 5 9999 9999 8 44 0 4 4 . 8 8-- 1— - 2 8 9999 9999 168 5 5 16 9999 9999 10 2 9999 9999 0 11 1 2 4 2 2 8 8 2 2 0 0

------------j -9---- 0-----&---- Q-----2-----0---- 0---- 1-----4-----2---- 4---- P-9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999

....... 9999 9999— 9999 9999. 9999 9999-9999-9999. 9999 9999-9999 99999999 9999 9999 9999

-AO^UHUL-ATED-A-W-A-Y-NES-S-;-^--------------------------------------------- --------8 8 8 10 10 10 10 8 8 10 10 10

— 0---- 0----- 4 ---- 4---- 6_____6_____6_____4— __ 4— -- 4---- 6 _ 66 6 8 8 10 10 10 10 8 8 8 100 0 4 4 6 6 6 4 4 4 4 6

-4----- 6-----6---- 8----1-0--- 10--- 10--- 10-----8-----8--- 1-0-- 1-0—10 10 0 0 4 4 6 6 6 6 8 88 10 8 8 8 8 10 10 10 0 0 4

-4 __ 6_____6 ____ 4____ 4_____4_____6_____8____ 6____ 6--- £2 1210 10 8 8 10 10 10 10 12 12 i 2 1214 14 14 12 12 14 14 14 14 8 8 8_0----- 0-----0---- 0-----0-----0-----0-----0-----0-----S---- 0--— 0—o o o o o o o , o b o o oo o o o o o o o o o o n 0 0 0 0

CUBE ( 90 , 63 ) IS SELECTED,ALL CORE CUBES HAVE BEEN SEARCHED,AND THE m a x i m u m c u b e d i m e n s i o n IS J 5

A10 6

CUBE ( ..rucr.K.J..KLG-

156 ; 63_pn i-N-Tg

) HAS b e e n f o u n d f

8 2 4 2 2 2 2 0 5 3 4 49999 9999 17 17 9 9 14 12 9999 9999 9999 99994 4 1 2 0 2 ? 5 9 49999 9999 17 17 9 14 9 12 20 9999 9999 99999959 9999 9999 4 1 0 0 1 0 4 2 2

..... 9 - 9-5995L 0900 9000 0 0 0 0 14 7 9999 0090 8 44 0 4 4 8 8 1 2 9 9 9 9 9 9999 168 5 5 17 9999 9999 14 2 9999 9999 0 13 __ 3 4 fl 9999 9999 999 9. .9.999. .. 2 ... 2 0 00 0 0 0 3 2 2 3 3 2 5 2

9999 9999 9999 9999 9999 9999 9999 9999 9999 9 9 9 9 9999 9999.... .9990-9999 0900 0900 0009 0 0 0 0 0999 9009 0000 0000 0000 .90909999 9999 9999 9 9 9 9 9999 9999 9999 9999 9999 9 9 9 9 9999 90909999 9999 0999 9 9 9 9

a c c u m u l a t e d a w a y n e s s J14 14 1 4 1 4 14 j 4 14 12 1? 1? 1 2 ...J.?0 O 6 8 8 8 8 6 4 4 6 66 6 14 14 14 14 14 14 12 1? 12 1?0 0 8 fl 8 8 8 6 6 4 4 66 6 6 16 16 16 16 16 14 14 14 14

14 14 0 0 4 4 8 8 6 6 16 1616 16 14 14 14 14 14 14 0 0 in10 10 1.0 fl 4 4 8 12 6 6 18 IB12 12 10 10 10 10 10 10 18 18 18 181R -I fl 1 fl 1 6 16 16 1 6 1 6 1 6 1 2 12 1?0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 i 0 0 0 0 0 n0 ■0 fl ft. fl p ft * 0 n p0 0 0 0

___C4I-BE - ( 156 1 63 ) IS RFIrFCTFD.CUBE ( 38 * 3 > HAS BEEN FOUND.CUBE ( 38 , 5 ) HAS BEEN FOUND.

r w a t c h .■FOR----- STAND­____ 5 ---------------- -.-.— -----

CHECKING POINTS10 4 6 9990 9009 0000 9090 n 9 R 6 — 6

9999 9999 17 17 11 11 16 12 9 9 9 9 9999 9999 99999999 9999 6 6 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 2 5 9 69099 OQ99 1 7 17 11 16 11 1? 2 0 9 00 Q_95 05_5 9999999 9999 9999 4 3 2 2 3 g 4 2 2

9 9 9999 9999 9999 9999 14 7 9999 9999 8 4-----4---- 1 ..........A . . ..4 ....fl.. .. 8 - ....... 1 ... 2 . --- 9-5999- 9999-- 16

A10 7

8 5 5 17 9999 9999 14 2 9999 9999 0 13 3 4 0 9999 9999 9999 9999 2 2 0 n

.__ - ........... 2 ..2 ..2 ....Q_____3 ... 2 ... .2 .3.... 3 ....2. - 5 29999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999..... 99.99. ..9.9 99 99-9-9 -.9-99.9. 929 9-999.9.-9 9 9 9 - -99.9-9.-99 9 9 - 9 9 9 9 - 9 9 9 9 - 99999999 9999 9999 9999

- .a c c u m u l a t e d AW AYNESS t• .... ... -.16 16 16 14 14 14 14 16 16 14 14 14n n 1? 1 2 1 fl 1 0 10 12 4 4 A A6 6 16 16 14 14 14 14 16 16 16 140 0 12 1? 10 10 10 12 12 4 4 66 6.. . . .. 6 ..... 20 18 18 18 .18 . 20 -.20 — IB. IP.

18 18 0 0 4 4 14 14 6 6 20 2020 18 20 20 20 20 18 18 18 0 0 161 6 14 14 __ 16. 4 4 14 1 8 6 6 22 2216 16 18 10 10 10 10 10 22 22 22 2220 20 20 22 22 20 20 20 20 IB is 18

..... .................................. 0 ... 0 . . 0 n ...0 0 0 0 ......................0. ................. 0 . . ... . 0 00 0 0 n 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 D n

...........................................................................- ..........................0. ............ . ..0 __________0 ..._____________0.

CUEE ( 3 8CUBE-CU.B£-

( 162(

CUBE ( 162CHECKING

51 ),1— 162 * 58 >_

is selected,HAS 0BEN f o u n d ,4±As -BEEN. .FOUND....

225 ) HAS BEEN FOUND.POINTS

10 6 8 9999 9999 9999 9999 9 9 9 9 —9 9 9 9 ___ 19___19.___ 11___ 11___ 1A_9999 9999 9999 9999 9999 9999 99999 9 9 9 9 9 9 9 9 9 9 9 9999 11 16 13

_? ?9 9_9999._9999____ 4___ 3. 2____9 9 9999 9999 9999 9999 146 1 4 4 8 8 1

10 5 5 17 9999 9999 143 3 4 8 9999 9999 99992 2 4 0 5 2 2

9999123722

99993

0 5 5 6.12— 9-999— 9999-9 9-99

220

____CL9999

99999

43

79999

4999999999999

43

119999__ *

89999

099999999

6-99-90-

69999

24

181

99999999

9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999J3 9 9 9 - 9 9 -9 -9—99 9-9— 9-9 9-9------------------------------------ — --------

ACCUMULATED AWAYNESS :

A108

200

— 6- 0 6

_26 22 18

—20- 24 0

- o . . .

0 0

180

— 606

.262220

-2-0 -

2400-00

1814-4-6-126

_.0 ..2620-24—220

__0_00

1414

-4-6-1226

..02622

-24-260

— 0_

0 0

14 16 -44- 14 26 _ 4_ 24 4

4-0-240

— 0-

0

3.414-14.1426...4244

4-0-260

_0~0

14144 4 .1224242622-40-260

— 0- 0

22181416242426224-0-240

- 4 30

204

-20-

1 6286

246

-24-240

— 0-

b

224

40-4

2 6606

-24-200

— 0 - 0

226

-48-4

28240

2618000

206

-206

28241826IPn0o

CUBE < 162 51 ) IS SELECTED,-CUBE < <4 , 123 ) HAS BEEN-FOUND , checking POINTS— ----------- 10---- 6-----8-99-99-99-99-999-9-9999 1 -7---- *>■— — 6 -- - 6

9999 9999 19 19 11 11 16 12 9999 9999 9999 99909999 9999 9999 9999 9999 99 99 9999 9999 9999 9 9999 P9999 9999 9999 9999 11 .16 13 14 9999 9999 9999 99999999 9999 9999 4 3 2 2 3 0 6 2 2

9 9 9999 9999 9999 9999 14 7 9999 9999 8 6_______________ 6-... 1____ -6—9-9-99—____10-.-9.9.9.9___1___ —4.___ 14— 99 9.9—9-9.99-.18

. 1 2 5 % 9999 9999 9999 14 2 9999 9999 0 13 3 4 8 9999 9999 9999 9999 4 4 9999 9999

. 2 2.. -4____ 0____5 ____ 2____ 2 — 3---- 3 ...- 3 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 0999 99999999 9999 9999 9999

A-CC U MU LATED--AW AAf'MESS— J24 24 22 14 14 14 14 24 22 28 26 260 n ?C1 1° 22 2? 22 . 4 4 ... _6— 66 6 16 16 14 14 14 14 20 20 10 220 0 12 12 20 18 16 18 16 4 4 66 __ 6.__..6. ___.3.2 34... 32 - 32 30 32. 28 ....-.34 32

32 3 0 0 0 4 4 30 28 6 6 28 2626 26 28 26 26 24 30 28 26 0 0 2220 26 24 29 4 A 26 P R 6 6 __3_4 -3?30 28 30 2R 10 10 10 10 30 28 22 2232 30 28 30 28 32 30 30 28 26 IB IB0 0 n n JJ 0 JD ‘ n 0 fl 0 - 00 0 0 0 0 0 0 0 b 0 0 n0 0 0 0 0 0 0 0 0 0 0 0

.... --0 ._..n.. n n

All

CUBE ( 41 , 123 ) IS 5ELECTED,CUBE ( 69 6 ) HAS b e e n f o u n d t • - -

cube ( 69 , 12 ) HAS b e e n f o l m d •

CUBE ( 69 , 70 ) HAS b e e n f o u n d .CHECKING POINTS

10 6 8 9999 9999 9999 9999 1 7 7 6 89999 9999 19 19 11 11 1 6 1 0 oaofi 0000 999-9A b -y vv9999 9999 9999 9999 9999 9999 9999 9999 9999 9 9999 89999 9999 9999 9999 11 16 13 14 9999 9999 9999 9999... 9999 9999 9999 6 9999 9999 9999 9999 .1. 8 9999 99999999 9999 9999 9999 9999 9999 16 9 9999 9999 8 66 2 6 9999 10 9999 2 6 13 9999 9999 IP

...— 12 ..— 5- ... 5 .....99.9.9 ...9.999 . 9999 .14 -.... 2..9999 9.999.--- 8. 13 3 4 B 9999 9999 9999 9999 4 4 9999 99992 2 4 0 5 2 2 3 3 3 9999 9999,--- — ... 9999 9999 9999 ..9999 99.99-9999 9999 9999 9999 9999 9999 9990

9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999

--------------------- -------- 9999 -99 99 9999- 9 9 9 9 --------------------------- ------------- ------------- ------------- ------

ACCUMULATED AWAYNESS •»

28 28 26 14 14 14 14 28 26 30 28 2fif 0 0 26f 24 26 26 24 28 4 4 6 6

6 6 1 6 16 14 14 1 4 1 4 2 n 26 1 8 260 0 12 12 26 24 22 26 16 4 4 66 6 6 34 34 32 32 30 34 30 34 32

...32. . ...30..._____ 0 . .___ 0 ____4 ... 4 ..32 ... 30 ___ 6 _ .— 6 __32-- 3030 28 32 26 30 24 32 30 28 0 0 2826 30 28 22 4 4 30 36 6 6 40 3836 34 38 36 in in 10 1 n . ._3B_ ... 36 ?2 ... 2238 36 34 38 36 38 36 36 34 36 18 IP0 0 0 n 0 o 0 0 0 0 0 n

.. 0 0 0 n 0 o 0 0 0 0 n 00 0 0 0 0 o 0 0 0 0 0 n

t 0 0 0 0

CUBE ( 69 • 78 ) IS SELECTED.CUBE < 164.., 225 )___HAS...BEE N. JOUNP .____ -..CHECKING POINTS____________ 1 n 6 8 9999 9999 9999 9999_____1 .7 7__ B___ -8 -

9999 9999 19 19 11 11 16 12 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9 9999 fl

,.... 9999 9999 9999. 9?99.__ 11___ 16___ 13___ 14 999.9 9999 9999 9999

A110

9999 9999 9999 6 9999 9999 9999 9999 1 fl 9999 99999999 9999 9999 9999 9999 9999 16 9 9999 9999 10 r 6---- 2_____6— 9.5IL9.9 ip 99 99____2_____£___13-9-9-99-5$ g_9___

12 5 5 9999 9999 9999 14 2 9999 9999 1 23 3 4 fl 9999 9999 9999 9999 9999 9999 9999 9999

.9 9 9 9 9 9 9 9 6 ____1___ 5 4 .... 4 3____ 3 . ... 4 9999 9 9 9 9

9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 999o9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 ..9.9.99 9999 9999 99 99 9999 9999 .999.9... 999.9. ...9.9.9.9 .9999 9999 9999 9999 9999 9999

..ACCUMULATED AWAYNESS32 34 32 14 14 14 14 34 34 36 34 360 0 .... .34 -.3? 3 7 34 3? 3 6 4- ___4____6--— 66 6 16 16 14 14 14 14 20 32 18 300 0 12 12 30 28 28 32 16 4 4 66 6 6 3P 34 32 32 30 40 38 34 3 7

32 30 0 0 4 4 40 38 6 A 34 3?34 32 36 26 36 24 36 34 34 0 0 3432 34 32 22 4 4 36 40 6 6 42 4n42 40 44 4? 10 10 10 10 38 36 22 2238 36 36 40 40 40 36 40 38 38 18 IPp 0 r, 0 n n 0 p £! 0 - 0-0 0 0 0 0 0 0 0 0 0 0 n0 0 0 0 0 0 0 0 0 0 0 n

- 0 ---0- -0* — a —

CyBECUBE

( 164

< 15q

225 )

31 >

IS SELECTED,HAS BEEN FOUND,

-GHE-G-K-In G -P-o-KNT-S- 10

9999 9999 9999 99999 9 9 9 9 9 9 9

9999 9999- 0 9 .9 -9 -—9999 -

6 212 5

..9-99.9 -9-994?-.9 9 9 9 9 9 9 9

9999 9999

821

.999999999999-99-99-

65

.4.6

9999

9 9 9 9 9 9 9 9

21 13. 99-9-9 9.9-99 9999 11

6 9999 -999-0— 9.999- 9999 109999 9999 B-_9_9-9_9-

1 59 9 9 9 9 9 9 9

9999 9999 9999.

16 9999 _99-0 9-

9999 19999 129999 999.9.

15 149999 9999 16---- 9-

99999999.9.9-99-

49999

2 614 2

.iL9-99.-__99-9-9_ 4 3

9999 9999

7 79999 9999 9999 - 99999 9999

1 8 -9005-9-9-99-13 9999

9999 9999 _9-9-99— 9-9-99

3 49999 9999

89999999999999999-- 1-0-

ln9 9 9 9

899999 9 9 9

--------------- s . .

99991

-9-9-99-09999999

9999 9999 9999 9999

9999 9999 9999 9999 9999 9999 9999 9999

9999 9999 9999

lfl2

999999999999

-9- 9999

ACCUMULATED AWAYNESS : — ------- 38--- 38 36 -14--- 14--- 1-4----14--- 4 8--- 30--- 40--- 3 8 -38

Alll

0 0 36 34 34 34 32 40 4 4 6 ft6 6 16 16 14 14 14 14 20 30 18 36- .... 0 0 12 12 34 32 30 38 16 4 4 ft6 6 6 4ft 34 32 32 30 48 44 34 3232 30 0 0 4 4 44 42 6 6 44 42

42 38 Aft 2ft .. 44 24- 4 4 —4 2 4.f) n A n*■ .* 1 ".......'t38 40 38 22 4 4 42 44 6 6

—.-- -48

--- *r46

42 40 48 46 10 10 10 10 38 36 22 22- 36 _ . 40 ..48 ..46 .-...46 . ...44 . . ..44 .... .4 2 _.44 18 18

0 0 0 0 0 0 0 0 0 0 0Ur Un

0 0 0 0 0 0 0 0 0 0 0 nn 0 n f) 0 _ 0-...... 0... -.. 0 __0 ..._Q .0 ---n.0 0 0 0

___ C-uBE___( . 1 5 0 * _31 -0___ ..IS .SELECTED. .. .- -...- ....

CUBE ( 174 . 189 ) HAS BEEN FOUND »CHECKING POINTS

10 6 8 9999 9999 9999 9999 1 7 7 8 10..... .. .._...9999. 9999 ....21. 21 . 13. 9999 _?9.99 . 12 9999 9990. 9999 9090

9999 9999 9999 9999 9999 9999 9999 9999 9990 0 9099 109999 9999 9999 9999 11 16 15 14 9999 9999 9999 9990ooop 9000 9999 ft 90 9 9_9. 9 -999 0.900 i-____fi .9009 99 9 09999 9999 9999 9999 9999 9999 16 9 9999 9999 10 fi

6 2 6 9990 10 9999 2 6 13 9999 9999 18...... ..T . 12. -.. 5 .. „ 5 .9999 .9999 9 9.9.9 ... .14 ___ 2 9999 9990 1 2

T 9999 9999 ;« 8 9999 9999 9999 9999 9909 9990 9999 99999999 9999 8 2 7 9999 9999 9999 9999 4 9999 90900090 0099 0099 0900 9090 0009 9099 ..99J9.9 -99.99 _9.&9-9 9.9.99 __9.99 9 -9999 9999 9999 9999 9999 9999 9999 9999 9999 9099 9999 99909999 9999 0999 9999 9999 9999 9999 9999 9900 9999 9999 99999999 9999 9999 9999

a c c u m u l a t e d a w a y n e s s f*46 46 44 14 14 14 14 46 44 44 42 420 0 44 42 40 34 32 46 4 4 6 66 6 16 16 1 4 1 4 14 1 4 20 4? 18. — .380 0 12 12 38 36 34 42 16 4 4 ft6 6 6 56 34 32 32 30 56 52 34 32

*2 3n 0 n 4 A «5 n 48 ft ft 52- 5950 44 52 26 50 24 48 46

t*'"44 0 0 46

46 46 44 22 4 4 46 50 6 6 54 5242 40 .. -5 .2 SO­ .. .10 10 -.— 10- 10 — 38- - 3 6 __22 ... 2738 36 42 SO 48 46 44

--ifc-W-—44 42 48 18 18

0 0 0 0 0 0 0 ’ 0 0 0 0 0o n n fl 0 n 0 (i ft 0 0 - .. fl0 0 0 0 0 0 0 0

. . — 0 0 0 f!

0 0 0 0

All 2

CUBE ( 174 . 189 ) IS SELECTED.C4JB-E (— 1-1-6__.— 49-- )--- H AS-!? E E-N—F-C-UNUD.-,--'--X u" . Tc h e c k i n g p o i n t s

$ W w .J-f.

1 n __ 6. 8 ...9 99 9 9999. 9.99.9. 9099 1 7 ... 7 fl 1 n9999 9999 21 21 15 9999 9999 12 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9 9999 109099 9999 S 3 9 S . 9-99 9 9 0 00 0009 1 7 1 fi 0000 -9 000. 0 o 0 0 00009999 9999 9999 6 9999 9999 9999 9999 1 8 9999 99999999 9999 9999 9999 9999 9999 16 9 9990 9999 12 10

--- 6 ...-2 ... 6. 9999 10 9995 ... _2 .. .6 ..13 _999.9 99.99. 2014 9999 9999 9999 9999 9999 16 2 9999 9999 1 2

9999 9999 4 8 9999 9999 9999 •9999 9999 9999 9999 9999OpQp qooo fl ? 7 0000 9090 0099 9009 fl 0909 90909999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9 99.SL_99.9_9 .9 999 9999 9999 9999 9999 99999999 9999 9999 9999

AfHIMill ATPn AU A VK'PCO 9

5 0 52 50 14 14 14 14 52 52 50 48 5oh-— ..............a 0 48 46 42 34 32 .... 50 4 4 6 6

6 6 16 16 14 14 14 14 20 48 18 420 0 12 12 38 36 36 44 16 4 4 66 6 6 _ .6JL 34 32 3.2 _ 30 69 60 34 . . _32._.

1 32 30 0 * 0 4 4 54 52 6 6 54 5254 48 56 26 56 24 52 50 50 0 0 50

................ 48 ... .46 44 22 4 .4 48 . 54 -6 ..... 6 . 60 __5842 40 58 56 10 10 10 10 38 36 22 2238 36 48 56 56 46 44 44 42 50 IB IPn n 0 0 n 0 0 0 0 0 n n0 0 O' n 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 n0 0 0 0

CUBE ( 116 , 49 > IS SELECTED'.CUBE ( 2 > 23 ) HAS BEEN FOUND,

--- CUBE... 1 2.., 39 ).. .h a s .. SEEN- FOUND . -- . - .......... - ---- — .CUBE ( 2 » 51 ) HAS BEEN' FOUND,WATCH FOR 23 AND 5lCHECKJNG PPJNTS

12 9999 9999 9999 9999 9999 9999 1 9 7 8 1fl9999 9999 9999 9999 15 9999 9999 12 9999 9999 9999 9999

A113

9999 9999 9999 9999 9999 9999 66-99 999-9-

9999999999996-955-

9999 9999 9999 9999 9999 9999

6 9999 9999 599 9999 9999-9

9999 9999 9999 11 999919 16 9999 9999 9999

9 9 9 9 9 9 9 9 ^ a 9999 1-6 __ 9 9999 -9-9-9-9—

9999

9999999*7999999999999

8 2 16 9999

-959 9— 9669- 9999 99999999999999999999

9999999999999999

69999

89999999999999999

9999 10 99999999 9999 9999

2 79 9 9 9 9 9 9 9

99999999 9999 9999 9999

9999999999999999

2 6 13 9999 999916 2 9999 9999 1

6566-6 966-6996-6696- 9999 9999 9999 5 90999999 9999 9999 9999 90999999-9999-9999 9999 9999 9999 9999 9999 9999 9999

ACCUMULATES AWAYNESS :5206

— 0-6

32.5.6...5042

- 3 6

0000

5206

_0-6

30-5-24640-3600

.00

-50.4816-1-2-60

.62.4464-5-2-00

14-4616-12-640

-26,2262-6-3-

-144614-3-8-344

-6.0-4

10- 4 6 -

143414-36-324

..__24 4

10 — 46-

000

00„Q.„

-.143214

- 3 . 8 _

32 62 -60 56 10 -4 4.00--0

565414-4-8-30 60 .55. 58 10 -44- 0 0

.0

- -54 4

20— 1-6-

68 6

5 6

638

— 4-2- 0 0

0

56 - 54 544 6 650 18 48

— 4- - 4— .664 34 3?6 58 56

........0 - .............. . 0 5?6 64 6236 22 22

-54-- 18 .180 0 n0 0 n

. .. 0 - - . 0 0

-C-UBE ( IS-SELECTED,-CUBE ( 170 50 ) HAS BEEN FOUND,CHECKING pOINTS

12 9999 9999 9999 9999 9999 ----— _9-9-9-9— 9-9-9 9— 9-9-96— 9.9-9-9---- 15-9-99-9

9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999

....... 999.9— 9 999 999.9____ 6.-99.9.9-999.99999 9999 9999 9999 9999 9999

10 2 6 9999 12 9999__________________ 1A Q665_ Q 9 gg Q 99Q 9000 Q Q 09

9999 1 9 7 8-&S-9-9— _12— 9-9 99— 9-9-99— 99 99-9999 9999 9999 9999 9999

19 18 9999 9999 999999.99 9999___ i____ 8 9999

16 9 9999 9999 122 6 13 9999 9999

16____ 2— 9666— 9-966---- 1-

10 -9 999

10 9999 9999

10 24

___299999999999999999999

9 9 9 9 9 9 9 9 4 8 9 9 9 99 9 9 9 9 9 9 9 0 3 9 9 9 9-6966—6566— 966.9— 9696— 6656-9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99999 9999 9999 9999 9999opop 9960 QQQO 0999______

9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 6 9 9 9 96 5 6 9 --9 5 5 6 -_ 6 „ 9 9 6 _ 9 6 6 5 —6 5 6 9 6 9999 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

ACCUMULATED AWAYNESS :

A114

-5806

520

. 6504816

144616

1454 1 4

14 34 1 4

14 32 1 4

6258

___14_5 0 30

. . 7 D 66 64 IP

584

— 2-0— 16 74

. ... 6- 62 6

38

624

5062 6

1 8586

---- -

06

. 32 ... 58 54

... 42--

06

.305846

.4 n....

126

...06644...fi-B.

1270n

26 22

.6fi_

3834

___.4_62 4

1 0

3632

___4_24 4

j n

4232

_7 0..66 62 1 fl

470

...6....D6

36

434620

6899

6326.25468

38 36 56 64 60 46 44 44 4 2 56~ r- £t-18 180 C 0 P 0 0 0 0 n 0 0 n

. P n 0 . _ D 0 0 0 0 0 0 n0 0 0 0 0 0 0 0 0 n 0 n0 0 0 0

CUBE ( 170 , 50 ) IS s e l e c t e d ,

C-U3-E____C__i 2__ » — 6-9___)__ HAS_B£EJv_£UU^!JU__c u b e ( 12 , 70 ) HAS BEEN FOUND,WATCH FOR 69 AND 70

— CUBE ( 12 . 93 ) - HAS BEEN FOUND,-.-CUBE ( 12 , 94 > HAS BEEN FOUND,CHECKING rQIN'tS 9

' 12 9999 9999» 9999 9999 9999 9999 2 9 9999 9999 1?...... 9999 9999 9999 9999 17 9999 9999 14 9999 9999 9999 9999

9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 1?9999 9999 9999 9999 9999 9999 19 18 9999 9999 9999 9999

_____-------9999- 9999 .99-9-9_____6._9_99-9~99SL9-_9j9-9_9_9-9-9-9____ ?____ £-_999-9-99999 9 9 9 9 9 9 9 9999,9999 9999 9999 9999 9999 9999 9999 12 10

10 2 6 9999 12 9999 3 7 13 9999 9999 24 16 99 99. 999.9..999.9... 9.999 9 9 9 . 9 ___ 1.6____ 2...999 9 -99-99 .1 2

9999 9999 4 8 9999 9999 9999 9999 9999 9999 9999 99999 9 9 9 9 9 9 9 e 3 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 6 9 9 9 9 9 9 9 9

___________ <?9<?9 9999 .9999 0999 999? 9909 9999 9999 9qq9 9999.9999— 99-90 _9 9 9 9 9999 0999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999

_________ 9.999 999.9 .9999. 99.99.____ - ._.... -ACCUMULATED AWAYNESS :

62 52 50 14 14 14 14 64 62 62 62 600 0 48 46 56 34 32 60 4 4 6 6

6.....6. 16 16 . 14.... 14___ 14___14. .. 20 50 10 560 0 12 12 38 36 48 54 16 4 4 66 6 6 74 34 32 32 30 76 74 34 32

... J32 3D 0_____0_____A_____4___10___ 7-0__ 6_____6--- 68 68

A115

66 64 70 26 68 24 68 68 66 0 0 6?62 46 44 22 4 4 64 70 6 6 74 7442. .4 0... 72 7? .10___ IP . 10 .10. .. 30 .36 22 2238 36 64 7 0 60 46 44 44 42 64 18 180 0 0 0 0 0 0 0 0 0 0 0n n 0 n n 0 0 _o. .0 _n. ... ..._0_ _.fL0 0 0 0 0 0 0 0 0 0 0 n0 0 0 0

CUBE ( 12 . 93 ) I S SELECTEDr,&JB.E I 2 5 , 55__ )___ HAS— BEEM-.FDU.Ng,

CUBE ( 75 . 127 ) HAS b e e n f o u n d .w a t c h FOR 59 AND 127rucrKTWR oniNr?

12 9999 9999 9999 9999 9999 9999 3 9999 9999 9999 129999 9999 9999 9999 17 9999 9999 14 9999 9999 9999 99999999 99 95. 599.9 _.5.9.99_9.959. .99 99. 5.9.95 9959 .5999 9999 9 9 9 9 129999 9999 9999 9999 9999 9999 19 18 9999 9990 9999 09909999 9999 9999 6 9999 9999 9999 9999 2 9999 9999 99909999 9995 99.9.9 .999.9 9990 0909 959.9 55.99-_ 9.95.9...55.99.___12-__ in-

10 2 6 9999 14 9999 3 7 15 9999 9999 2418 9999 9999 9999 9999 9999 18 2 9999 9990 1 2

9999 9999 4 p. 9999 9999 9999 ..9.999 -5999---99 90. 5099 5.95.9999 9999 ■o8 3 9999 9999 9999 9999 9999 6 9999 99999999 9999 9959 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999

ACCUMULATED a w a y n e s s

66 52 50 14 a 4 14 1 4 66 6 6 6 "» 62 -- 64.......... w ft

0 0 48 46 62 34 32 64 4 4 6 66 6 16 16 14 14 14 14 20 50 18 6n

____ 0 D 12 12 38 36 52 58 16 4 4 66 6 6 80 34 32 32 30 00 74 34 3?

32 30 0 0 4 4 70 70 6 6 74 727n 68 74 26 70 24 74 70 6 fl n 0 — 66— vD64 46 44 22 4 4 70 76 6 6 62 8042 40 78 76 10 10 10 10 38 36 22 22

.. - ... -. — 38 -— 36-__7-0--— 75- 60 ---4 6_.4.4 ...44- -. ,_4 2-— 7 2 .10 180 0 0 0 0 0 0 0 0 0 0 00 D 0 0 0 0 0 0 0 0 0 0n 0 p 0 n p n n * n 0 • 8 -

0 0 0 0---CUBE ( 75 .59 -)..IS -SELECTED,--- ..

All6

CUBE < 98 • 5e ) HAS BEEN FOUND,CHECKING pOINrS

12 9999 9999 9999 9999 9999 9999 3 9999 9999 9999 1? ..... 9999 9999-9999-9999...17 9999 99-99 14 9999 9990 9999 9999

9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 129999 9999 9999 9999 9999 9999 19 20 9999 9999 9999 9999 ______0-99-9-99-9-9-99-99_____ 6-999 9-9-9 99-99-99-9-9-9 9 ? 99-9 0— 9 999 -9-9 9 Q-9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 14 In9999 2 7 9999 9999 9999 3 7 15 9999 9999 9999

20 9999 9999 9999 9999 9999 -18 2 9999 9999 1 29999 9999 A 8 9999 9999 9999 9999 999? 9999 9999 99999999 9999 8 3 9999 9999 9999 9999 9999 6 9999 9999

___________ 999 9_9 9 9Q—9 9 9 9— 99-9 9— 9 9-9 9— 9 999— 9 9 9-9---999 9 —999-9—9-999— “99 9 -999 09999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999 9 9 9 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999

______ ____ 9999 9999 9999 9999 ............ACCUMULATED AWAYNESS :

72 52 50 14 14 14 14 72 62 62 62 680 n 48 46 70 34 32 68 4 4 6 66 6 ... 16.. __1.6.. .14.„_1.4._ .14. -14 .20 ._. 50 18 660 0 12 12 38 36 56 60 16 4 4 66 6 6 84 34 32 32 30 84 74 34 32

3n 0 n 4 4 .7.D 70.._. .6 6 -7-6 _ .7670 7? 76 26 70 24 78 78 72 0 0 66

f 66 46 44* 22 4 4 74 84 6 6 88 8842 4.0 B..4f... R 4 _10 1 0 .. .10 .. 10 ...38-- 36 . .22 .2238 36 76 80 60 46 44 44 42 76 18 180 0 0 0 0 0 0 0 0 0 0 0n n n n p n 0. . .0- .-0- . . fl -0— fl-0 0 0 0 0 0 0 0 0 fl 0 n0 0 (3 0

CUBE < 98 1 58 > is s e l e c t e d 9

4-- 45- -24__ )----wXS-8£c h e c k i n g p o i n t s

1 4 9955_S?99_9595.. 9999 9999 __4_ 999.9-9-9.99.9999 9999 9999 9999 9999 9999 9999 14 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 21 20 9999 99999999 9999 9999 6 9999 9999 9999 9999 2 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9990

____________0090 9 7 0990 9999 999_9_____3_____B___ 15-4-9-9.9.20

9999 -9-9 99-

9999 9999 -9-9 99-

99994

____ 8-

9999fi

___ 3-99999999-9L949-

99999999-9.99-9-

1899999990

29999.9999.

99999999

-9-9-99-9999 9999 ____6

9 9 $ 9 - - - . 17- 9999 9999 9999 9999 9999 9999 9999 9999

14 10_ 9 9 9 9 - 9 9 9 9

1 2 9999 9999 _9909-9999

A117

9999 9999 9999 9999 9999 99999999 9999 99999999 9999 9999

9999 9999 9999 9999 9999 99999999 999 9 .9 9999999

9999 9999 9999 9999 9999 9999 9999 9999 9999

9999 9999 9999 9999 9999 9999 9999 9999 9999

-A-GCUW-UL A7ED-A WA YA'ESS74 52 50 14 14 14 14 74 62 6? 62 7?0 ... 0 . 48 ..46 70 _34 32 72 4 4 6 66 6 16 16 14 14 14 14 20 50 IB 6 60 0 12 1 2 38 36 58 64 16 4 4 66 6 .... ..............6 9n 34 32 3 0 9 p — 7-4 34 3?

32 30 0 (} 4 4 70 70 6 6 82 8070 76 62 26 70 24 82. 80 76 0 0 66

._72 — 46 ....... 4 4 . .2.2 .__4 _ __ 4__ 78 _ _8 8. . _ 6 _ 6 . 94 9? .....4 2 40 90 88 10 10 10 10 38 3 6 22 2238 36 80 86 60 46 44 44 42 8 2 18 180 0 0 n 0 n 0 o fl ft 0 n0 0 0 0 0 0 0 0 n 0 0 n0 0 0 0 0 0 0 0 0 0 0 o0.. ... 0 ...0 . . n . . ____ ................. ................ ... ........... ..... ..................

CUBE ( 45 i 21 ) IS SELECTED,CUBE C 14 . 66 ) HAS BEEN' FOUND,

CUBE < .14 92 J ___ HAS-BEEN-F-OUblB-,WATCH FOR 1 68 AND 9 2

CHECKING POINTS14 9999 9999 9999 9999 9999 9999 4 9999 9999 9999 9999

_ _ 9999 9999 9 9 9 9 9999 9 959 99-59 9999 14 9990 .9-99-9 -9.99.9- 9999 ..9999 9999 999.9 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 21 20 9999 9999 9999 99999 Q 9 9 0999 0999 7 9090 0000 9099 0990 ? 9-99.9-..9.99-9 9999 -9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 14 m9999 2 7 9999 9999 9999 4 8 15 9999 9999 9999

20 9999 9999 9999 9999 9999 18 2 9999 9990 ... 1. — 29999 9999 4 8 9999 9999 9999 9999 9999 9990 9999 99999999 9999 8 3 9999 9999 9999 9999 9999 6 9999 99990 0 9 0 0999 9099 0900 9090 0000 9099 9999 9000 _9j95-9_ 0009 9099.-- .9999 9999 9999 9999 9999 9999 9999 9999 9999 9990 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999

ACCUMULATED AWAYNESS J 178 52 50 14 14 14 14 78 62 62 62 7?0 0 48 46 70 34 32 76 4 4 6 66 6 16 16 ..14 14. 14 _.14_ _20..50.. .is 66

A U8

0632

06

3n126n

1292n

38344

36324

643270

7030-70

16 92 6 -

474

...64

34.86

632

----866670 80 86 26 70 24 84 84 80 0 0

80 46 44 22 4 4 82 96 6 6 98 9 84 2 _ 40 _ 96 96 _L0. _ _ 10 10__ _3fl___36.__ -22 2238 36 86 92 60 46 44 44 42 9 0 IB 18

— 0 0 0 0 0 0 0 0 0 0 0 n0 0 0 c 0 0 0 0 n 0 0 n

— 0 0 0 0 0 0 0 0 0 0 0 n0 0 0 0

CUBE < 14 , 68 > IS SELECTED,-G-UB E---- (-- 2rA-r— -5-c---- 1--RA5—gEE — P-OTJ-W -,CUBe ( 24 f 62 ) HAS BEEN ^0LJD.WATCH pOR 50 AND 62G-HEGX-I-NG--POW-S---------- — ----------------------------------------------

14 9999 9999 9999 9999 9999 9999 4 9999 9999 9999 99999 9 9 9 9999 <3999 9999 9999 9999 9999 9999 9999 9999 9999 9999

— ........ -9999 999-9- 999-9— 9999 9999- 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 21 9999 9999 9999 9999 99999999 9999 9999 7 9999 9999 9999 9999 3 9999 9999 9999

--------- f 9-9 q_-9 -9 9 9 9-99— 9-999- 9999----1-4----100999 2 17 9999 9999 9999 4 8 15 9999 9999 9999

20 9999 9999 9990 9999 9999 20 2 9999 9999 1 2_________ 5999 9999____ 5____ £ 9999 9999.9-999 -9999 9999 9999-9999 9999

9999 9999 8 3 9999 9999 9999 9999 9999 6 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999

----------- 9-9fl$_9J9-9-9— 9J99-9— 9-95-9-7-9-99— 99-99— 999-9— 9-999— 9999—9999— 9999 -9999 -9 9 9 9 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999

ACCUMULATED AWAYNESS S-8-4----------5-2- -59- 44- 4 4 ___ 14- 44-06

... 0 6

32 -78- 86 42 -38 0 0

-43 -

0 6

... 0.

6 30

- 88-

46 40 36 0 0

— 0 -

48 4616 1612 126 980 0

-943--- 26-44 229 8 1 0 0■94 -------960 00 0

— 0_____________0-

7014.3.8.3444

10-60-00

- - 0-

3414.3.6_ .324

-78------ 24-

321468 .

327 0

-8-8-

4 10

-46 0 0

__0-

84 10 -4 4 0 0

_ 0-

-82-7614-7.0. 30 70 90-102 ‘ 10 — 4 4 -

0 0

0-

. 85 ------62-4 4

20-16.966

-86-

6 38 -4 2- 0 0

— 8-

50.4.746

— 0-

636-94-00

— O -

-62 6

18 _ 4 34 92 — 0-104

221800

-- 0

-7?6

666

3294

— 6610622IBnn

_ ,-n

A 1 1 S

CUBE ( 24 . 62 ) IS SELECTED.

CUBE ( l , 23 > HAS BEEN FOUND.CHECKING p o i n t s

9999 9999 9999 9999 9999 9999 9999 5 9999 9999 9999 9990r 9999 .999-9...9.999 9 9.9 9 _9_9 99 9999 .9 -9-9.9 9999 9999 9999 9999- 99999999 9999 «999 9999 9999 9999 9999 9999 9999 9999 9999 9999

9999 9999 9999 9999 9999 9999 23 9999 9999 9999 9999 99999090 0000 0099 7 9099 0009 9099 9999 * 9000 0 0□0 9QQo9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 14 m9999 2 7 9999 9999 9999 4 8 15 9999 9999 9999

........ - .2 D .9999 9 9 9 9 -99.99 9999 9999 ...... 20 ..- 3 -99.99. 9999 - 1 39999 9999 5 8 9999 9999 9999 9999 9999 9999 9999 99999999 9999 8 3 9999 9999 9999 9999 9999 6 9999 99909090 0goo 0090 Q990_ 90 90 9009 9099 9 9 9 9 - 0009 goo© 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99909 999 9999 9999 9999

ACCUMULAT ED AWAYNESS :64 52 50 14 14 14 14 84 62 62 62 720 0 48 46 70 34 32 76 4 4 6 6fi .6. 16 1A 1 4 1 4 14 1 4 ?n 5fl 1 80 0 12 12 38 36 70 70 16 4 4 fi6 6 6 102 34 32 32 30 102 74 34 32- 3? 3n f? f! 4 4 70 70 ft A Qfl O r

70 92 96 26 70 24 96 96 92 n 0 6fi90 46 44 22 4 4 92 104 6 6 108 106-4.2...- --4.0 - .10 4 1.0 4 1 0 i n 10 1 P 38 -36 2-2 2238 36 98 104 60 46 44 44 42 100 18 IB0 0 0 0 0 0 0 0 0 0 0 0n o •0 n n n p D A n p .... . fl ..

0 0 0 0 0 0 0 0 0 0 0 n0 0 0 n

c u b e < 1 1 23 > IS SELECTED,

GU££----(-1-1-1— r— 9 ±---1--- H-A-S-B E E-N—F OU-NBCHEc K i mG POi mTs----------- 9 9 9 9 — 9-9-99— 9-9-9-9— 9-95.9_9-9-9-9_9-9J9-9 SJ9-SL9 5 -9-9-99 999 9 -9999 9999

9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999

-9-99-9— 9-9-99 -9-9-9-9 -9999 99-9-9— 9999_______________ 2.3— 9 9 99 - 9 9 9 9 - 9 9 9 9 - 9 9 9 9 - 9 990 _

9999 9999 9999 7 9999 9999 9999 99?9 3 9999 9999 99909999 9999 9999 9999 9999 9999 9999 9999 9999 9999 14 IP

._____ 9999 - -3 7 9959 9 9 9 9 -9999____ 4____ 9 _9999 9999 9999 999922 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 20 3 9 9 9 9 9 9 9 9 1 3

A120

9999 9999 5 6 9999 9999 9999 9999 9 9 9 9 9999 9999 99999999 9 9 9 9 -

9999fl * 9999 9-999- ...9999 9999-

99999 9 9 99999

-99999999

999999999999 9999 9999 9999 9999 9999 9999

9999 9999 9999 9999 9999 9999 9999 9999 99 99 9999 9999 9999.......... ........... .. 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999

9999 9999 9999 9999

A r PI IMIII ATE 11 AIJAVMECQ • ' ‘ ' ~84 52 50 14 14 14 14 88 62 62 62 7?

0 ._ 0..... 48 ....46 .... 7 0. 34 32 ......76 ....... 4 -.. . 4 . 6 . . 66 6 16 16 14 14 14 14 20 50 18 660 0 12 12 38 36 74 70 16 4 4 66 6 6 108 34 32 32 30 106 74 34 . 3 2

10232 30 0 0 4 4 70 70 6 6 10470 94 102 26 70 24 100 98 92 n 0 66

.......... ........................ 92 .. 46 ....4.4 . . . 2.2. 4 4 96 ..11.2. ____ 6-____ 6 ..118 .11642 40 112 110 10 10 10 10 38 36 22 2?38 36 104 112 60 46 44 44 42 110 18 18n n n 0 0 n n n 0 p 0 00 0 0 0 0 0 0 0 0 n 0 n0 0 0 0 0 0 0 0 0 n 0 n

........ 0. ...0 _... 0 ....0 .CUBE ( 111 , 91 ) IS SELECTED,CUBE ( 124 , 30 ) HAS BEEN FOUND,

. . C H E C K I N G P O I N T S A. . __________________ ____________________________ ________________________9999 9999 9999 9999 9999 9999 9999 5 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9 9 9 0 0090 0999 9090 9909 9099 0900 ooog 9000 9999 9.9.999999 9999 9999 9999 9999 9999 23 9999 9999 9999 9999 99999999 9999 9999. 7 9999 9999 9999 9999 3 9999 9999 99909999..999? 9999 9 99? . ?. ? 9 ? 9999 9999 .999?..9.9.94..9999.._. -14 109999 3 7 9999 9999 9999 ’5 9 9999 9999 9999 9999

22 9999 9999 9999 9999 9999 9999 3 9999 9999 1 39999 9999 5 8 9999 9999 9999 9999 9999 9999 9999 99999999 9999 8 3 9999 9999 9999 9999 9999 6 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999-9-9-9 -9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999

ACCUMULATED a w a y n e s s : .. 04 52 50 14.. 1 4 . - 1 4 14 - 94 ■ 62 62...62.- 7 2

0 0 48 46 70 34 32 76 4 4 6 66 6 16 16 14 14 14 14 20 50 IB 66

-------------- 0----- 0----15--- 1-2-- 38----36----7-8--- 7-0--- -1-6— — 4 4 — 66 6 6 114 34 32 32 30 110 74 34 3?

A121

32 3070 10098 4642 4030 -360 00 0

• 0 -00 c

010644

116-14-2-

00

00

0 4 4 70 70 6 6 110 lin26 70 -24 102 102. 92 n 0 6622 4 4 96 120 6 6 126 126116 10 10 10 10 38 36 22 22418----60 — 46. -.44-.— 44 — — 4 ? - 116 -4-8 — 18---0 0 0 0 0 0 n 0 Cl0 0 0 0 0 0 0 0 n. _ 0 -........- 0 .... __13 -.....-....0 . . . . 0 0 0 0 n . .

-C- U B E -f-1-E4- - 3 0 — )- - - - - - - o s — s -e w e c -t-e -d - . -

CUBE ( u s , 48 > HAS BEEN FOUND.

CUBE ( 115 , 57 ) HAS BEEN FOUND,-W-A-7-GH—F-Qr----- 4 A— AND---- 57-----------------CUBE ( 115 » 58 ) HAS BEEN FOUND.WATCH FOR 48 AND 58-C44E CK TNG ..-PO-I-ivT-S-

99999999

99999999 f 9999 -9-999-

9999999999999999999999-99

39999

9999 9999 .9 99-9.-9999 9 9 9 9 9 9 9 9

9999 9999 9990 9999

9999999999999999999$-9995?

79999___ 5_

8999'99999

9999 9999 9 999. 9999

7999999999999____2

3"9999S-9-9-9.

999999999999999999999 9 9 99999 9999 .99 99 9999 9999 -99JSL9-

9999 9999 .99 99. 9999 9999

9999 9999 .9 999

25 9999

99 9-9— 99-99-9999 9999 .99 99 9999 9999 .9.999.

5999999.99.99999999_9999_

599999 9 9 . 999999999-9999

93

999-9..999999999999-

9999999999999999

3-999999999999-9.9-9999999999

9999 9999 9999 9999

99999999

99999999

9999 9999 9999 9999

9999 9999 9090 9999 9999 .9999 9999 9999 9999 99999990--- 159990 9999 9999 19999 9999

7 9999 9999 9999

9-9-9-9— 9-9-9-9— 999 9. 9999 9999 9999

9999 9099 9999 9999 9999 — 11 9999

3.9999 9999 9999 -9.999 - 9999

ACCUMULATED AWAYNESS :84 5 2 5n 1 4 1 4 14 1 4 1 no 62 69 62 720 0 48 46 70 34 32 76 4 4 6 66 6 16 16 14 14 14 14 20 50 18 660 n l? 1 2 38 36 80 70 1 6 4 4 ...66 6 6 118 34 32 32 30 116 74 34 32

32 30 0 0 4 4 70 70 6 6 ii2 11?7 p 104 11 n ?6 7 n 24 1 08 1 p 8 p 0 .6698 46 44 22 4 4 96 124 6 6 132 13242 40 122 122 10 10 10 10 38 36 22 2?38 36 118 124 60 46 44 44 4? 118 IB IP

A122

0 0 0 0 0 0 0 0 0 n 0- 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0T] 0 0 0 0

CURE ( 11-5- 48 J JS RFLFCTFP'.CUBE ( 55 » 3 ) HAS BEEN FOUND.

c u b e ( 55 > 5 ) HAS BEEN FOUND,ia.TCH.rOa -3 AND 5..........CUBE ( 55 . 17 ) HAS BEEN FOUND,

WATCH FOR 5 AND 17

-CUBE... C 55 , 27 ) -HAS BEEN. pOUND...

WATCH FOR ± 7 AND 27

C U B E ( 55 . 29 > HAS B E E N FOUND,

W.A.I-G-H -E-OR.....17.-And.... - —2 9 -------------C U B E < 55 , 57 ) HAS B E E N F O U N D ,

WATCH FOR.- ± 7 AND £ 71

CHECKING POINTS .. ............... - . -. ....I 9 9 9 9 9999 9999 9999 9999 9999 9999 5 9 99 9 9999 9999 9999j 99 99 9999 9999 9999 9 9 99 9999 9999 9999 9 9 9 9 9999 9999 9999I________________ 9-999—9S9 9 -9 9 99—9-9-9S—99-99--9.9-99—9S9S—9.999-999-9—99S-9-9-999....9 999

9999 9999 9999 9990 9 9 99 9999 9999 9999 9 9 9 9 9999 9999 99999909 9999 9999 7 9 9 99 9999 9999 9999 3 9999 9999 9999

„............ .......... 9999. 9999 9999 9999 9 9 99 9999 9999. 9999. 9999 9.999 15 1 19 99 9 3 7 9999 9 99 9 9999 5 9 9999 9999 9999 99999999 9999 9999 9999 9 9 9 9 9999 9999 4 9999 9999 1 3

_________________959-9—9-9.99._____ 5_____g _ 5 5 5 5 —9555-.95-95—9 .9 9 5 -5 5 9 5 —9555_5959. -.9 9.9 99 9 9 9 9999 8 3 9999 9999 9999 9999 9 9 9 9 7 9999 99999999 9999 9 9 9 9 9999 9999 9999 9999 9999 9 9 9 9 9999 9999 9993

9 9 9 9 9999 9 999 9999 9 9 9 9 9999 9999 9999 9 9 90 9999 9 9 9 9 99999999 9999 9999 9 9 9 9 9999 9999 9999 9 9 9 9 9 9 9 9 9999 9999 99999 9 9 9 9999 9999 9999

ACCUMULATED AWAYNESS 5 8 4. 52__50 _ 14 14 14 14 104___ 62.. ...62___ 62......72

0 0 48 46 70 34 32 76 4 4 6 66 6 16 16 14 14 14 14 20 50 18 66

_______________0.____ 0___ 12___ 12___ M__3.fi___ 8.0___ 7-0___ 16____ 4---- 4--- 6.

A123

6 6 6 124 34 32 32 30 124 74 34 3?32 3 C 0 0 4 4 70 70 6 6 110 11670 _1G0 118 26 __7.0 24 _ H 6 l 11.4 .......92... 0 .. _..o .. 6698 46 44 22 4 4 96 126 6 6 138 13642 40 128 126 10 10 10 10 38 36 22 2?38 36 122 132 60 46 44 44 49 ... .12.2- ie.__ JL80 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 n 0 n0 0 0___ 0_____0 .P..___ 0 0 0 .... n . 0 c0 0 0 0

nnRP ( RR 1 7 ) IS RPLPDTFD,

CUBE ( 120 161 ) HAS BEEN FOUND .

CUBE < 128 , 224 ) HAS BEEN FOUND.CHECKING POINTS

9999 9999 9999 9999 9999 9999 9999 5 9999 9999 9999 99909999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999909 9090 0909 0990 9999 0999 9099 0999 0990 9999 9.955 9 9 9 09999 9999 9999 9990 9999 9999 9999 9999 9999 9999 9 9 9 9 99909999 9999 9999 7 9999 9999 9999 9999 3 9999 9999 99999 9 95— 55 95— 955-9— 5955— 9555— 5555— 5955— 955 9-555 9— 5 9 99 ---15 - 419999 3 7 9999 9999 9999 5 9 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 5 9999 9999 9999 9999

-- - ... . - 9999 9999 5 ..8 9999 9599 99 99 9999 9999 9 9 9 9 9999 99999999 9999 P 4 9999 9999 9999 9999 9999 8 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999

a 995—99£KJ_995-9-99-95-9559— 9595— 9559-5599— 9999 9999 9999-99999999 9999 9999 9999 9999 9990 9 9 9 9 9999 9999 9999 9999 9999 9999 9999 9999 9999

ACCUMULATED AWAYNESS : .--------------64---- 55---5-0----14--- 14----14----14--10B--- 6-2---6-2--- 62---7 ?

0 0 48 46 70 34 32 76 4 4 6 66 6 16 16 14 14 14 14 20 50 10 66

_______________0_____ 0___12___ 12____38___ 36___ 80___ 70___ 16____ 4 ...._. 4 66 6 6 128 34 32 32 30 130 74 34 3?

32 30 0 0 4 4 70 70 6 6 122 120--------------7-9---116--124--- 2-6----75--- 24-- 1-24-- 125--- 92----4-----0---66

98 46 44 22 4 4 96 128 6 6 130 13642 40 132 130 10 10 10 10 38 36 22 22

.38 -— 3 6 __ 126 —134___ 60 .. .46— 44 __ 4-4- — 42- - 125 . — IB 180 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0

— n----------0-------------0------------ 0------- — p— . . . 0 --------0 — L _ 0 _ ---------0 --------- 0 - ---------0 — 00 0 0 0

r — CUBE ( 128 , 161 ) IS SELECTED,

A124

9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 129999 9999 9999 9999 9999 9999 6 9 9999 9999 9909 99909999 .0900^0 00 9000 °999 9999 9999 .. 5 -_9Q_9o_0000 Oooo Ooon9999 9999 5 0 9999 9999 9999 9999 9999 9999 9909 99999999 9999 9 4 9999 9999 9999 9999 9999 8 9999 9999

_______ .5999 99 99 99.9.9...9 999 J9.9 9.9 99.99 .9.999 9 9 9 9 -99.9.9-- 9 9 9 9 . 99.99. 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9990 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99990090 .0 9 90 0 0 90 _9955

ACCUMULATED AWAYNESS :84 52 50 14 14 14 14 118 62 62 62 7?0 0 40 46 70 34 32 76 4 4 6 6<4 (4 1 A 1 6 1 4 1 4 14 .. 14.. -2-0- . .5.0. — 3.8. 660 0 12 12 38 36 80 70 16 4 4 66 6 6 134 34 32 32 30 138 74 34 3?

32 ... 30. 0 p 4. 4 70 70 ...6 .... 6 .128 12670 116 132 26 70 24 132 130 92 0 0 6698 46 44 22 4 4 96 144 6 6 138 1364? 4.0 150 14 R 1 n 1 0 10 -1.0-..38- ._36 . 22 2?36 36 138 148 60 46 44 44 42 140 18 180 0 0 0 0 D 0 0 0 0 0 n

.... 0 .. 0 ... 0 0 ___ 0 ___ 0......0... .. 0 . 0 n 0 0D 0 0 0 0 0 0 0 fi 0 0 00 0 0 0

CUBE ( 72 . 96 ~f) , IS SELECTED.

...CUBE JL l6S . j.1-6.2- HAS_BE£N._FOUJJEL. -----

CUBE ( 168 164 ) HAS b e e n f o u n d

WATCH FOR 162 AND 164CUBE..... ( 168 i 174.. ) HAS BEEN .FOUND,WATCH FOR 162 AND 174CHECKING POINTS

9690 . 999° 9999 9999 9999

IP 9999 -9999 9999 9999

-9999 - 9 999 -9999-9999—9999— 9999— 9999-9999 -9999— 9999— 9999 9999

-9999-9' 9999 9999 9999 9999

59

9999 9999 9999 5 99999999 9999 9999 9999 99999999 9999 9999 9999 99999999 9999 9999 9999 99999999 9999 9999 9999 99999999 9999 9999 9999 99999999 9999 6 9 99999999 9999 9999 9999 99999999 9999 9999 9999 9999

9999999999999990999099999999

9999,909999999999999999999999

9 9999

A125

CUBE ( 103 . 79 ) HAS been f0lnd.----WATCH FOR .. 7 -AND . .79 ... -..— ... ... . -.

CHECKING POINTS---- 99-99 99 9-9 -9999 -3-9-99--9-99-9— 9-9 99— 9933- 5- 9-99-9--9939 -9099 9999-

9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999____ 99.99— 9999 .9.9 99 -99.99 -9.9 9.9— 99 9.9 9.99.9. 99.99 - 9 9 9 9 9999 3999 99999999 9999 9999 7 9999 9999 9999 9999 3 9999 9999 9999

9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 15 110909 ppoo 7 0900 9000 0 9 99 «5 0 QOOo 0090 OOOO Oflfin9999 9999 9999 9999 9999 9999 9999 5 9999 9999 9999 99999999 9999 5 P 9999 9999 9999 9999 9999 9999 9999 9999

_ ..9.9.99_ -9.995.... ...9 4 99? 9_ 993_9 99?9 9999 .9999 .. 8 9999 99909999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999909 0000 009 0 _9..939 .9999.3399 933 9 0 9 0 0 0006 QOOO -909 9_ 90009999 9999 9999 9999 - y ~ v V

___.ACCUMULATED .AWAYNESS •- •. -84 52 50 14 14 14 14 114 62 6? 62 72p n 4 ft 46 2.0 -34 32 ___76. ...4.___ 4..____6 ...£6 6 16 16 14 14 14 14 20 50 18 660 0 12 12 38 36 60 70 16 4 4 66 6 6 134 34 32 32 30 130 74 34 3?

32 30 0 0 4 4 70 70 6 6 128 124' 70 116 I3f 26 70 24 130 126 92 0 0 66

98 46 44 22 4 4 96 136 6 6 138 136A 2 40 144 14 0 10 10 10 10 38 36 22 2236 36 12B 144 60 46 44 44 42 134 18 180 n 0 n n n 0 n h n n G0 0 0 0 0 0 0 0 0 n 0 n0 0 0 0 0 0 0 0 0 0 0 nn n n 11

CUBE ( 103 , 7 ) IS s e l e c t e d ,CUBE ( 72 » 66 ) HAS BEEN FOUND..0UB£— -X-- 72-- , — 96-- )---H Ag__BE Etv.-F-OLjsjD-,CUBE ( 72 , 108 ) HAS BEEN FOUND.CHECKING POINTS

9999 9999 9999 9999 9999 9999 9999 5 999? 9999 9999 9999----------- 999 9-39 9 9-39 93-9333— 9-933— 99-93--3333— 9939— 9993— 99-93-3339-9 999

9 9 9 9 9999 9999 9 9 9 9 9999 9999 9999 9999 9999 9999 9999 99999 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

—— ________ 9.9 9.9 9999 .. 9.39SL.99J.9.SSJ9 -9 .9 9.9._9_9.9 9 .3 9 .9 3 ...9 9-9.9 - 9 0 99.. 9999 9 99 9

A 126

9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999........................

ACCUMULATED AWAYNESS 584 52 50 14 14 14 14 124 62 6? 62 720 0 48 4 6 70 34 32 76 4 4 6 6- .— .. ..... 6 . - 6. .16 -.16 .14 14 14 ..14 20 50 18 660 0 12 12 38 36 80 70 16 4 4 66 6 6 134 34 32 32 30 138 74 34 32

y, 2 30 0 p 4 4 70 70 * 6 — 1-28- 1-3?--70 116 132 26 70 24 138 138 92 0 0 6698 46 44 22 4 4 96 . 150 6 6 138 1364 ? 40 .-15 4...

142154 1 n 10 - . . -10- _ 10 — 38—

42-36 • --22

182?lfl38 36 148 60 46 44 44 142

0 0 0 0 0 0 0 0 0 0 0 nn 0 0 n n n 0 0 0 0 0 ---n

nC 0 0 0 0 0 0 0 0 0 00 0 0 0

CUBE ( 168 , 162 ) IS SELECTED,-CUBE--- (— — 9 ,---- 3--)-— HA-5— BE-E-Ki—F-0 UND-,CUBE < 9 » 5 ) HAS BEEN FOUND,WATCH FOR r 3 AND 0

1E 4---9— — -43— )----H AS-fi-E-EK-F- -U-NXU-

WATCH FOR 5 AND 43c u b e < 9 , 45 ) • h a s b e e n f o u n d .4W-A-T-G H-F-O-P----- 5_4 N D---- 45-----------------CHECKING POINTS

9999 9999 0999 9999 9999 9999 9999 99999999 9999 9999 9999 5955 95.55_

999999999999

99999999_99-9.9-

999999995559-

999999999-95-9-

9999999955.95

9999 9999 9 999

9999999999995959-

9999 99999999 9999 9999 9999 _9.555_5555

9 9 9 9 9 9 9 9 9 9 9 9 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999

-_5.955 -9599-.59.95— 9.955— 9595-9955 6____ 5_9999 9999 9999 9999 9999 9999 9999 59999 9999 5 8 9999 9999 9999 99990999 9500 9 0990 9090 9995 9 9 9 9 9 995

9999 9999 9999 9999 9999 9999 5-959 5999-5595 9999 9999 9999 9999 9999 9999 5555_____9-5555-

9999 9999 9999 9999_9999 999.99999 9999

99999999

99999 9 9 9

5555-999999995555.

999999995555

99999999-9599

99999999.9555-

99999999.59-99.

9999 9999 9999 9999 5555-5559

9999 9999 9999 .555 9_ 9999

1? 9999 9999 9999 5550- 9999 9999 9999

9 9 9 9 9 9 9 9

A12 7

.. __ ....

a r r 11m111 A T c n i u i V H F c ? t

8 4 5 2 5 0 1 4 1 4 1 4 1 4 1 2 4 62 6 2 6 2 7 2

0 0 4 8 4 6 7 0 3 4 3 2 7 6 . . . 4 . . .4 6 6

6 6 1 6 1 6 1 4 1 4 1 4 1 4 20 5 0 18 66C 0 12 12 3 8 3 6 8 0 7 0 1 6 4 4 66 6 6 1 3 4 3 4 3 2 3 2 3 0 1 3 8 7 4 3 4 3 ?

3 2 3 0 0 0 4 4 7 0 7 0 6 6 1 2 8 1 3 67 0 1 1 6 1 3 2 2 6 7 0 2 4 1 4 4 1 4 2 9 2 0 0 669 f l 4 A 4 4 2 ? 4 4 9 6 1 5 4 ......._ 6 . .. . 6 —1 3 8 . 1 3 64 2 4 0 160 1 5 8 10 10 10 10 3 8 3 6 22 223 8 3 6 1 4 8 1 4 8 6 0 4 6 4 4 4 4 4 2 1 5 0 1 8 1 6

n n 0 0 0 f) n._ n . _R- n .0 n

0 0 0 n 0 0 0' 0 0 0 0 n0 0 0 n 0 0 0 0 0 n 0 n

--------------- ---------- ---- ....0 .. . . D __ .... 0 ....n - ....... - ■ --- - ...

CUEE < 9 , 5 ) IS SELECTED.CUBE < I 45 * 1 ) HAS BEEN FOUND,

- .CHECKING POINTS9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99999 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 0 9 6 9 9 9 0 9 0

...9 .9 9 9 . 9 9 9 9 . 0 9 9 9 9 9 9 9 . 99.9.9.. 9 9 9 9 . . .9 .9 .9 9 -9 -9 9 .9 __9 9 9 9 - 9 9 - 9 9 ...9 9 9 9 . .9.9999 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

9 9 9 9 9 9 9 9 0 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9

1 9 9 9 9 9 9 9 9 9 9 9 $ _ 9 9 9 9 9 _ 9 9 9 9 9 9 9 . 9 9 9 9 9 9 0 9 0 9 9 .9 9 9 .9.9 .9 .9 9 .9 .. 1 2

9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 6 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 90 0 9 0 OQOO _ 5 0 9 0 0 0 9 0 0 9 9 9 9 9 . 9 9 9 9 - 9 9 9 9 9 9 9 9 . - 9 - 9 .9 9 . . 9.9999 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 0 9 9 9 9 9 9 9 9

9 9 9 9 9 9 9 9 - 0 9 9 9 . .9 9 9 .9 . 9 .9 .9 9 .. J 9 . 9 9 9 9 9 9 .9 9 9 9 9 . 9 9 9 9 - 9 . 9 9 9 . 9 9 9 9 . 9 9 9 9

9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 0 9 9 9 9 9

9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

ACCUMULATED AWAYNESS •*84 ......5 2 5 0 . 1 4 . ....... 1 4 ........1 4 1 4 1 2 4 ..... 6 2 . 6 2 6 2 7 2

0 0 4 8 4 6 7 0 3 4 3 2 7 6 4 4 6 6

6 6 1 6 1 6 1 4 1 4 1 4 1 4 2 0 5 0 1 8 6 6n n 1 2 1 2 3 8 3 6 8 0 7 0 . 1 6 4 4 ...... 66 6 6 1 3 4 3 4 3 2 3 2 3 0 1 3 8 7 4 3 4 3 ?

3 2 3 0 0 0 4 4 7 0 7 0 6 6 1 2 8 1 4 0

7 Q 1 1 6 1 3 2 ..... .26 . 7 0 2 4 1 5 4 1 5 0 .......9 . 2 ._G_...0. . 6 6

9 6 4 6 4 4 2 2 4 4 9 6 1 5 4 6 6 1 3 8 1 3 6

4 2 4 0 1 6 4 1 6 0 1 0 1 0 1 0 1 0 3 8 36 2 2 9.7

. . 3 8 36__ISA. .1.4 fl_.. . 6JL.-.46.. ......4 4 . . . ......4 4 ... 42 -.-154-____ 1 8 . ...IB

A128

0 0 0 0 0 0 0 0 n 0 0 00 0 0 0 0 0 0 0 n 0 0 00 ... 0

0..0

. 0 0

00___ 0.. 0.. 0 .0 .0 0 0 0

-CLlSE L-145- -1 1 _I_S_S ELE-C.T-EJO-*.CUBE ( 152 28 >c u b e ( 152 157 )-OUfiE____ ( 152__3— 217_WATCH FOR 157 AND 217

HAS BEEN FOUND,

HAS "BEEN FOUND, has BEEN FOUND,

CUBE ( 152 i ■ CHECKING POINTS.

220 ) HAS BEEN FOUND.

9999 9999 9 99 9 9 9 99 9999 9999 9999 9999

9999 9999 9999 9 999 9999 9999 9999 9999

9 99 99999

.999999999999999999999999

99999999.999999999999999999999999

9 99 9 9999<9999 9999 jp.9 9 9 .-9 !9999 9999 9999 9999

9 9999 9999, 9999

0999 9999 9999 9999

99999999999999999999999999999999

99999999

99999999999999999999999999999999

99999999

999999999999999999999999

69999

99999999

999999999999999999999999

99999

99999999

99999999999999999999999999999999

99999999

9999 9999 9999 9999 9999

9999 9999 9999 9999 9999 9999 9999 9999 999o 9999 9999 9999 9999 9999 9999 9999 9999 1 ?9999 9999 9999 9999 9999 9999

-9-9-9-9—9-9-99—9 9 9 9- 9 9999 9 9 9 °

9999 9999 9999 ' - 9 9 9 9

9999 9999 9999

ACCUMULATED AWAYNESS-840

— 520

— 5-0 48

1-4—46

— 14 70

— 1434

-- 1-432

124-76

---62-4

6 2 -4

— 62 - 6

- 7 26

6 6 16 16 14 14 14 14 20 50 18 66-0 .12 ...12 - -.38.-36- __8 .0 -..7-0- 16- . . 4 - -4 66 6 6 134 34 32 32 30 138 74 34 3?

32 30 0 0 4 4 70 70 6 6 128 14B-7-0— 116— 132--- -2-4--- 7-0— — 2-4— 16-2— 15 8---9-2---- B---- 0----6ft98 46 44 22 4 4 96 154 6 6 138 13642 40 164 160 10 10 10 10 38 36 22 2?3 8 ____ 36___1.6.4__ 14.8____ 6.0___ -4 6 ____ .44____ 4 4____-4-2__15 H_16— 1 B

0 0 0 0 0 0 0 0 0 0 0 0o o o o o o o o b o o o

_0_____ 0-----0____ 0_____ 0____ 0_____0____ iO_____0____ 0---- 0---- n0 0 0 0

}— cube c 152 ; 217 i is s e l ec te d,

A129

CUBE ( 106 , 72 ) HAS BEEN FOUND •CUBE ( 108 . 77 ) has b e e n FOUND •

--- CUBE ( 108 , 92 ) HAS b e e n FOUND.c u b e ( 108 , 105 ) HAS BEEN FOUND «w a t c h FOR 77 AND 105

r-— CHECKING PC I NTS9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999_999j?_9999--999-9--9-9-99— 9999 --9-9 99-99-9-9--9-99-9--9999--9999--099-9--99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99909999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999_ _____9999. 9999 -999.9- 9 9.9 9 .999 9 .9999 -9999..-9 9-99-.9999 -9999 999 9 139999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999OOOO OOOQ QOOO 0900 90QQ 9090 0900 09C50 OOOQ OQOO OOOO -9990 ..9999 9999 9 9999 9999 9999 9999 9999 9999 9 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999_______ 9999 -99 99- 9 9 9 9 9-99 9 9999.-9999 .9.99 9 9999 .9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999

- 9999 9999 9999 9999ACCUMULATED AWAYNESS ••

... '.. ...84. 5 2 . sJl __ 1 A.___1.4. 1.4........14 124 ... 62 ..62. 62 720 0 48 46 70 34 32 76 4 4 6 66 6 16 16 14 14 14 14 20 50 ie 66n n 1? 12 3 8 36 80 7 0 18 A 4 ____6 ..6 6 6 134 34 32 32 30 138 74 34 3?

32 30 ,0 0 4 A 70 70 6 6 128 15070 116 132 26 70 24 162 158 9? P 0 6698 46 44 22 4 4 96 154 6 6 138 13642 4 0 164 160 10 10 10 10 38 36 22 2?38 36 172 148 60 46 44 44 42 166 18 18C 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0p fi n n n n n n 0 n 0 . . .0 -0 0 0 0

---CUg£ .< 10-8--- —1-0-5— -4--- 4-S_S£-L£-C4-£D-,--- --

CUBE ( 176 , 50 ) HAS BEEN' FOUND,CUBE < 176 , 162 ) HAS BEE* FOUND,

A 1 3 0

c h e c k i n g p o i n t sr 9999 9999 9999 9999 9999 9999 9999 9999 9999 5999 9999 9999

9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9909 99909999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999

-------------- 9 999 9 9-9 9 9 9 9 S —9SL9JB—9 9 9 .9 —99 9 9 -59-9-9...9-9.9 9—9 999— 99-9-9- 9999_ 9.999.9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 13

_______ _______9 9 99.9999 .999.9 9999 9.9 99 ...9.99.9. .99.99 999.9.9 9.99 9.999 ...9 999 9 9909999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999

-------------- .999-9— 999-9---- 9— 9-999.._999-9-9 9 9 9 ^ 9 9 9 9 —9 9 9 9 - 9 9 99 -9 9-99 Q Q 99-99 go.9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999

- 9999 9999 9999 -99.99 9999 9595. 9999 9999 -9999 9999 9999 99909 9 0 9 9 9 9 9 9 9 9 9 9999

-A CCU K'-U LATE D- AW A Y-N ESS i — — -----------------------------------------------84 52 50 14 14 14 14 124 62 62 62 7?0 ...0 ..48 46 .70 34 .. 32. 76 ...4 ...... .. 4 6 66 6 16 16 14 14 14 14 20 50 18 660 0 12 12 38 36 80 70 16 4 4 6

_ 6.___ 6. 6 134 34 32 __ 3 2. __3.O.-_13.8. __74-. .34 .... 3232 30 0 0 4 4 70 70 6 6 126 15670 116 132 26 70 24 162 158 92 0 0 6698 .46 .. . .. 44... 22 4 4 96 154 ... 6 .. .. ..6. .138 13642 40 164 160 10 10 10 10 38 36 22 2?38 36 17*8 148 60 46 44 44 42 166 18 18n 0 >n ft o ft 0 ft 0 n ft n0 0 0 0 0 0 0 0 0 0 0 n0 0 0 0 0 0 0 0 0 0 0 00 .... .0 . .0 __0. ..

c u b e ( 176 , so >• is s e l e c t e d ;

c u b e ( 167 , 35 ) HAS BEEN FOUND, CUBE- i - 1.62 ____,_3.7.__L___ HAS— B EE.N.. F_Q UND.,

WATCH FOR 35 AND 37

CUBE ( 167 . 161 ) HAS BEEN FOUND, Watch .for 37 A ^n__1.61__________________

CUBE < 167 , 173 ) h a s BEEN FOUND,WATCH FOR 37 AND 173 CHECKING POINTS

A131

9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99990000 OOOO qooo 0000 Onoo 0000 OoQQ 9995 _955_6 OOOO Qooo A *79999 9999 9999 9999 9999 9999 9999 9999

' “ -y-y-9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999--------------9-95-9-5555— 9-959--5955—9-955--9959-.59-95--9999-5999--9999--9959--9959-

9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99509999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99909999 9999 9999 9999

ACCUMULATED AWAYNESS *•- - _ _ ...... 84 . -52 .5.0. ..14 ... 14 14 14 .124 62 62 62 7?

0 0 48 46 70 34 32 76 4 4 6 66 6 16 16 14 14 14 14 20 50 18 66O O 1? 12 30 36 SD 7-0 ... 1.4 -4-_______4 . A

6 6 6—A c. -134 34 3? 32

--— r~V

30«■» w

138 74 34......

3?32 30 0 e 4 4 70 70 6 6 126 160

..... ... -... 70 --116--- 132--.26 70 .24 - -162 ...-158 92 ___0- 0 6698 46 44 22 4 4 96 154 6 6 138 13642 40 164 160 10 10 10 10 38 36 22 2230 36 178 14 8 6 0 46 44 44 4 9 166 1-8 1 Au u ....^ ...— 4. v w

1 0 0 & 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 n

— ... - -... -...0.. ..... 0 .. ....0 - - 13 . — 0- ___0 -..0 ______.0 --- 0 - _______0--..0 n0 0 0 0

CUBE < - W -- »-- 37-- )----I-S-SE-L-E-C-T-ECCUBE ( 97 , 104 ) HAS BEEN FOUND.CUBE ( 97 , 112 ) HAS BEEN FOUND,

H-A-T-CH FOR— 104-AND 11-2----------------CUBE ( 97 . 121 ) HAS BEEN FOUND,

watch for 112 and 121cube---<— 9 7 — ,— 254— >-----h-a-S-bheai -fc un-d -

WATCH FOR 112 AND 224CHECKING POINTS

9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999

A132

------------ ------- 9999 9 9 9 9 - 9 9 9 9 - 9999 -9.999 • 9999. 999.9 9999- 9999 -9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9 99 9 9999 9999 9999 9999 99915 " 9 9 9999

. 9 9 9 9 - 9 9 5 9 .99 99. 9999 9 9 9 9 9999 .9999 9999. 9999 9999 ...9999 99999999 9999 9999 9999 9 9 99 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9990

-------------------- 99-99—99-99—" 9 -9 —9-"-9—9-9-99—99-99—9-999—99L99_99-99 -9-9-9-0—9999—-9-9 9-Q9999 9999 9999 9999 9999 9999 9999 9999 9999 5999 9999 90999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999

....................... 9999 9999 9 9 9 9 . 9 9 9 9 9999- 9999 " 9 9 9999 9999 9999 9999 99999 9 9 9 9 9 9 9 9999 9 " 9 9 9 9 9 9999 9999 9999 9999 9999 9999 99999 9 9 9 9 9 9 9 9 9 9 9 9999 9999 9999 9999 9999 9999 9999 9999 9999

_____________ QOOO Q O O O 0 0 0 0 O9Q0_____________________________________________________

ACCUMULATED AWAYNESS :84 52 50 14 14 14 14 124 62 62 62 7?0 0 48 46 70 34 32 76 4 4 6 66 6 16 16 14 14 14 14 20 50 18 660 0 12 12 38 36 80 70 16 4 4 66 6 6 334 34 32 32 30 138 74 34 32

- _32___3.0. ___Q__ _ (i 4 4 70 .ZD . __ 6.. _ _6. IPS -1606670 116 132 26 70 24 162 158 92 0 0

98 46 44 22 4 4 96 154 6 6 138 1364? 4 n 1 64 160 1 0 in 1 0 10 38 36-.-- 22 -

18— 2?

IP* C.38 36 178 148 60 46 44 44 42 1660 0 0 0 0 0 0 0 0 0 0 0Q. - .0 v 0 .... 0 . .0 ... 0 . .0 0 0- - 0 . . 0 . n0 0 < 0 0 0 0 0 0 6 0 0 00 0 0 0

CUBE ( 97 , 224 ) IS SELECTED.

^♦FORTRAN ## STOPyU^riTiF--------------------------- -------- ------------

C B 4 2 Q LOGOFF AT q 9 4 9 ON 11^12 ' * f'OR T 5 N 36q6,C E421 CPU TIME USED; 0Ol3eB.966O SECONDS.

A133

APPENDIX I

EXAMPLE OF 9 VARIABLES, 5 ON-SET MINTERMS, AND 50 3 DON'T CARE MINTERMS.

A134

TOATAL NUMBER OF ON-SET AND DONT«CARES = 508ON'SET c 5 D0NT-CARES = 503n u m b e r s OF v a r i a b l e s = 9

• m i n t r l 5 1 h i n t f'JU 2 = 2 m i n t tt 3 c 4MINT U 4 S 8 MINT it 5 3 256 MIN \tt 6 B 0MINT # 7 s 3 MINT tt 8 g 5 MIN , i 9 c 6MINT # 10 - 7 MINT 11 3 9 MIN ,* 12 B 10MINT 13 e 11 MINT tt 14 c 12 MIN ,tt 15 s 13MINT # 16 s 14 m i n t tt 17 3 15 MIN , ( t 18 s 17MINT 19 16 MINT tt 20 c 19 MIN 21 S 20

i MINT i 22 s 21 m i n t tt 23 3 22 MIN ,tt 24 B 23! MINT # 25 3 24 MINT tt 26 25 MIN .tt 27 26

MINT * 28 e 27 MINT #' 29 S 28 MIN ~,i 30 B 29MINT if 31 a 30 MINT it 32 3 31 MIN 33 *■ 33MINT n

i im 34 MINT tt 11 = 35 MIN 36 B 36

1 .. ............ m i n t # 3 37 MiNT tt = 38 M in .tt 39 B 39m i n t # 40 3 40 MINT tt 41 41 MIN .tt 42 B 42

! MINT #. 43 S 43 MINT tt 44 - 44 MIN .tt 45 B 45MINT # 46 « 46 MINT it 47 = 47 MIN .it 48 3 48MINT # 49 3 49 MINT tt 50 s 50 MIN , i 51 S 51m i Nt a 52 3 52 M int tt 53 s 53 m IN .tt 54 B 54

j " MINt tt 55 55 MINT it 56 s 56 MIN , i 57 B 571 MINT # 58 3 58 MINT tt 59 s 59 MIN .tt 60 S 60

MINT it 61 61 MINT i . 62 M_ 62 MIN 63 s 63MINT it 64 3 65 MINT tt 65 66 MIN .tt 66 B 67m i n t a 67 3 68 M I N t tt 68 s 69 MIN 69 3 70m i n t # 70 2 71 m i n t tt 71 s 72 MIN .# 72 B 73MINT m 73 3 74 * MINT tt 74

... __~ 7 5 “ MIN 75 C 76

MINT tt 76 «• 77 ’ MINT it 77 s 78 MIN .it 78 E 79MINT tt 79 3 80 m i n t tt 80 s 81 MIN .# 81 S 82MINT tt 82 3 83 m i n t tt 83 S 84'“ MIN Vtt 84 s 85MINT tt 85 3 86 MINT tt 66 S 87 MIN .# 87 - 88MINT tt 88 ■* 89 MINT tt 89 s 9 0 MIN .it 90 B 91MINT tt 91 *• 92 ‘ MINT it 92 s~ 93 MIN .a 93 S 94

, MINT # 94 «* 95 MINT tt 95 B 96 MIN r* 96 s 97MINT it 97 *■ 98 m i n t tt 98 s 99 MJN ,# 99 E 100MINT tt 100 3 _ 1Ql MINT tt 101 c 1 0 2 “ MIN V i 102 " B 103MINT # 103 3 104 MINT tt 104 s 105 MIN , i 105 s 106MINT # 106 g 107 MINT tt 107 s 108 MIN .it 108 s 109MINT 109 ? 110 MINT tt~110" E n r MIN ,i 111 B 112MINT # 112 5 113 m i n t it 113 ' - 114 MIN .tt 114 B 115m i n t # 115 116 MINT n 116 S 117 MIN .# 117 S 118M 1 NT 118 119 MI N r tt 119 C 120 MIN Vi 120 B 121MINT 121 m 122 MINT tt 122 S 123 MIN .tt 123 S 124MINT # 124 3 125 m i n t tt 125 B 126 MIN .tt 126 E 127m i n t # 127 3 129 MINT tt 128 ' B 130 MIN . i 129 B 131m i n t it 130 3 132 MINT tt 131 B 133 MIN , i 132 = 134MINT tt 133 3 135 MINT tt 134 = 136 MIN

MIN. i 135 B 137

140MINT tf 136 3 138 ' MI NT #“ 137 B 139 . i 138 SMINT it 139 3 141 MINT it 140 142 MIN .tt l 4 l B 143

A135

m i n t tt 142 S 144 MIN .tt 143 = 145 MIN .tt 144 s 146MINT tt 145 s 147 MIN .tt 146 E 148 MIN .tt 147 E 149MINT tt 148 E 150 MIN .tt 149 g 151 MIN .tt 150 3 152MINT tt 151 g 153 MIN .tt 152 3 154 MIN .it 153 B 155MINT tt 154 156 ' MIN 155' "s 157 “ 'MIN .tt 156 m 158m i n t tt 157 s 159 m i n ,tt 158 E 160 m i n .tt 159 = 161m i n t tt 160 s 162 MIN ,tt 161 Z 163 MIN ,tt 162 m 164MINT tt 163 s 165 ~MIN ,tt 164 ."g' “ 166 MIN .* 165 C 167MINT tt 166 JJ 168 MIN .tt 167 g 169 MIN .tt 168 g 170MINT tt 169 •» 171 MIN .tt l7o E 172 MIN .tt 1 n s 173m i n t tt 172 m 174 m i n ,tt 173 175 " M in \tt 174 z 176m i n t tt 175 z 177 MIN .tt 176 E 178 MIN .tt 177 E 179m i n t tt 178 s 180 MIN . tt 179 = 181 WIN .180 _E 182MINT tt 1 8 1 g 183 MIN '.it 182 g 184 MIN .tt 163 C 185MINT tt 184 s 1 8 6 MIN .tt 185 S 187 MIN .tt 1 8 6 E 1 8 8

m i n t tt 187 E 189 m i n ,tt 188 g 190 m i n .tt 189 E 191MINt tt 190 E 192 MIN ,tt 191 E 193 MIN .tt 192 E 194MINT tt 193 c 195 MIN ,tt 194 Z 196 MIN .tt 195 S 197MINT tt 196 198 MIN j 197 __S. 199 MIN j. tt. 198._E 2 0 0

MINT tt 199 s 2 0 1 MIN .tt 2 0 0 s 2 0 2 MIN .tt 2 0 1 C 203m i n t tt 2 0 2 s 204 m i n .tt 203 E 205 m i n .tt 204 5 206MJNjt tt 205 -5_2P7______ JT.I N ,tt 206 E 208 . MIN ,tt_ 2Q7_ __s„ 209MINT tt 208 s 2 1 0 MIN .tt 209 C 2 1 1 MIN .tt 2 1 0 E 2 1 2

MINT tt 2 1 1 s 213 MIN .tt 2 1 2 g 214 MIN .tt 213 E 215MINT tt 214 s 2 1 6 MIN .tt 215 E 217 MIN ,tt 2 1 6 E 2 1 8

M jNT tt 217 E 219 m i n ,tt 218 = 2 2 0 MIN .tt 219 E 2 2 1

M I Nt MINT

tt 2 2 0 g 2 2 2 MIN .tt 2 2 1 ■ £ 223 MIN .tt 2 2 2 E 224tt 223 z . 225 -MIN 224 m 226 225._E _ 227

MINTtf

226 228 y MIN .tt 227 229 MIN *tt 228 E 230MINT 229 s 231 , MIN ,tt 23o E 232 MIN .# 23 1 E 233M tNT tt 232 z 234 M IN .tt 233 Z 235 - m IN .tt 234 E 236HI Nt tt 235 s 237 MIN .tt 236 E 238 MIN .tt 237 E 239MINT tt 238 g 240 MIN .tt 239 S 241 MIN' .tt 240 S 242MINT M 241 243 MIN 242 _ g_ 244 MIN 243 245MINT tt 244 g 246 . MIN , tt 245 E 247 MIN .tt 246 E 248M INT tt 247 z 249 m i n .tt 248 = 250 m i n ,tt 249 m 251M I Nt tt 250 s 252 MINj .tt 251 p* 253 MIN ,tt 252 s 254MINT tt 253 z 255 MIN .tt 254 • g 257 MIN ;# “235 P 258MINT tt 256 - 259 MIN .tt 257 ' g 260 MIN .tt 258 261MINT tt 25?

262s 262 MIN .tt 260 :S 263 MIN .tt 2 6 1 S 264

m i n t ft z 265 MIN .tt 263 E 266 m in .tt 264 P 267m i n t tt 265 E 268 MIN ,tt 266 E 269 m i K . # 267 c 270MINT tt 2 6 8 z 271 MIN .tt 269 E 272 MIN .tt 270 = 273MINT tt 271 274 MIN ,tt 272 g 275 MIN .tt 273 B 276MINT tt 274 z 277 MIN ,tt 275 Z 278 MIN .it 276 F 279MINT tt 277 280 MIN .tt 278 z 281 MJN .it 279 B 282m i n t tt 2 8 0 g 283 MIN .tt 281 E 284 MIN .tt 282 B 285m i n t tt 283 g 286 MIN .tt 284 g 287 MIN .tt 285 B 288MINT tt 2 8 6 g 289 MIN ,tt 287 E 290 MIN .tt 288 S 291MINT tt 289 s 292 MIN ;tr 290 g 293 MIN 9 $ 291 B 294m i n t tt 292 3 295 m i n .tt 293 ' m 296 MIN .9 294 9 297MINT tt 295 = 296 MIN ,tt 296 E 299 HINTjJL 2 ?y_._.1__ 300

A13(

MIN .9 2 9 8 8 3 0 1 MIN ,9 2 9 9 S 3 0 2 MIN ,9 3 0 0 8 3 0 3MIN .9 3 0 1 s 3 0 4 MIN ,9 3 0 2 8 3 0 5 MIN .9 3 0 3 8 3 0 6MIN .9 3 0 4 5 3 0 7 MIN .9 3 0 5 S 3 0 8 MIN .9 3 0 6 8 3 0 9MIN .9 3 0 7 8 3 1 0 m i n .9 3 0 8 8 3 1 1 MIN .9 3 0 9 8 3 1 2MIN ,9 3 1 0 5 3 1 3 MIN ,9 3 1 1 C 3 1 4 MIN .9 3 1 2 8 3 1 5MIN ,9 3 1 3 •5 3 1 6 MIN ,9 3 1 4 g 3 1 7 MIN' .9 3 1 5 8 3 1 8MIN .9 3 1 6 S 3 1 9 MIN .9 3 1 7 8 3 2 0 MIN .9 3 1 8 g 3 2 1MIN .9 3 1 9 s 3 2 2 MIN ,9 3 2 0 8 3 2 3 MIN .9 3 2 i = 3 2 4MIN .9 3 2 2 s 3 2 5 m i n .9 3 2 3 8 3 2 6 m i n .9 3 2 4 g 3 2 7

^ ........... MIN ,9 3 2 5 g “ 3 2 8 MIN .9 3 2 6 = 3 2 9 MIN .9 3 2 7 g 3 3 0MIN .9 3 2 8 s 3 3 1 MIN * 9 3 2 9 g 3 3 2 MIN .9 3 3 0 C 3 3 3MIN .9 3 3 1 8 3 3 4 MIN .9 3 3 2 m 3 3 5 MIN .9 3 3 3 8 3 3 6MIN ,9 3 3 4 8 3 3 7 MIN .9 3 3 5 8 3 3 8 MIN .9 3 3 6 8 3 3 9MIN .9 3 3 7 C 3 4 0 Mi n ,9 3 3 8 8 3 4 1 m i n .9 3 3 9 •» 3 4 2MIN .9 3 4 0 S 3 4 3 MIN »9 3 4 1 8 3 4 4 MIN .9 3 4 2 8 3 4 5

' MIN . # 3 4 3 s" 3 4 6 MIN ,9 3 4 4 8 3 4 7 “ MIN .9 3 4 5 = 3 4 6[ MIN .9 3 4 6 g 3 4 9 MIN ,9 3 4 7 g 3 5 0 MIN .9 3 4 8 g 3 5 1

' MIN .9 3 4 9 s 3 5 2 MIN .9 3 5 0 g 3 5 3 MIN .9 3 5 1 8 3 5 4MJN V# 3 5 2 s 3 5 5 MIN ,9~ ' 3 5 3 8 “3 5 6 MIN .9 3 5 4 C 3 5 7MIN .9 3 5 5 s 3 5 8 MIN ,9 3 5 6 8 3 5 9 MIN .9 3 5 7 8 3 6 0MIN .9 3 5 8 - 3 6 1 MIN ,9 3 5 9 = 3 6 2 MIN .9 3 6 0 8 3 6 3MIN .9 3 6 1 m 3 6 4 MIN , 9 3 6 2 8 3 6 5 MIN .9 3 6 3 8 3 6 6MIN .9 3 6 4 - 3 6 7 MIN .9 3 6 5 8 3 6 8 MIN .9 3 6 6 8 3 6 9MIN .9 3 6 7 m 3 7 0 m i n , 9 3 6 8 = 3 7 1 - M I N ,9 3 6 9 3 7 2MIN .9 3 7 0 "" = " 3 7 3 MIN ,9 3 7 1 8 3 7 4 M.IN .9 3 7 2 8 3 7 5MIN .9 3 7 3 s 3 7 6 MIN .9 3 7 4 8 3 7 7 MIN .9 3 7 5 8 3 7 8MIN . 9 3 7 6 g 3 7 9 MIN .9 3 7 7 g 3 8 0 MIN ,9 3 7 8 8 3 8 1MIN 3 7 9 8 . 3 8 2 „ MIN .9 3 8 o 8 3 8 3 MIN .9 3 8 1 = 3 8 4m i n 3 8 2 S 3 8 5 , m i n ,9 3 8 3 8 3 8 6 m i n .9 3 8 4 8 3 8 7MIN .9 3 8 5 8 3 8 8 MIN ,9 3 8 6 = 3 8 9 MIN .9 3 8 7 = 3 9 0MIN .9 3 8 8 S 3 9 1 MIN .9 3 8 9 g 3 9 2 MIN .9 3 9 0 8 3 9 3MIN . 9 3 9 1 g 3 9 4 MIN . 9 3 9 2 g 3 9 5 MIN .9 3 9 3 8 3 9 6MIN , 9 3 9 4 g 3 9 7 MIN . 9 3 9 5 8 3 9 8 MIN ,9 3 9 6 8 3 9 9

r~ ' “ MIN , iT 3 9 7 8 4 0 0 , m i n ,9 “3 9 8 8 4 0 1 m i n .9 3 9 9 g ” 4 0 2MIN .9 4 0 0 8 4 0 3 MIN ,9 4 0 1 S 4 0 4 MIN ,9 4 0 2 4 0 5MIN .9 4 0 3 4 0 6 MIN .9 4 0 4 8 4 0 7 MIN .9 4 0 5 8 4 0 8MIN ,9 4 0 6 g 4 0 9 MIN .9 4 0 7 8 4 1 0 MIN .9 4 0 8 8 4 1 1MIN .9 4 0 9 4 1 2 MIN .9 4 1 0 8 4 1 3 MIN .9 4 1 1 P 4 1 4m i n .9 4 1 2 8 4 1 5 m i n ,9 4 1 3 g 4 1 6 M IN .9 4 1 4 8 4 1 7

, ......" MIN .9 4 1 5 8 ' 4 1 8 MIN .9 4 1 6 g 4 1 9 MIN V# 4 1 7 8 4 2 0MIN .9 4 1 8 q 4 2 1 MIN ,9 4 1 9 g 4 2 2 MIN .9 4 2 0 8 4 2 3MIN .9 4 2 1 s 4 2 4 MIN .9 4 2 2 8 4 2 5 MIN .-9 4 2 3 , - S ... 4 2 6MIN ,9 4 2 4 4 2 7 MIN ,9 4 2 5 8 4 2 8 MIN .9 4 2 6 p 4 2 9

m i n .9 4 2 7 8 4 3 0 m i n .9 4 2 8 8 4 3 1 m i n .9 4 2 9 8 4 3 2MIN .9 4 3 0 8 4 3 3 MIN ,9 4 3 1 8 4 3 4 MIN .9 4 3 2 8 4 3 5

^ “ MIN .9 4 3 3 c’ " 4 3 6 MIN ,9 4 3 4 F*» 4 3 7 MIN .9 4 3 5 8 4 3 8MIN • 9 4 3 6 8 4 3 9 MIN .9 4 3 7 g 4 4 0 MIN .9 4 3 8 8 4 4 1MIN .9 4 3 9 - S 4 4 2 MIN ,9 4 4 0 8 4 4 3 MIN ,9 4 4 1 s 4 4 4

m i n ,9 4 4 2 8 4 4 5 m i n ,9 4 4 3 8 4 4 6 m i n .9 4 4 4 m 4 47

MIN .9 4 4 5 8 4 4 8 MIN i 9 4 4 6 g 4 4 9 MIN .9 4 4 7 8 4 5 0MIN .9 4 4 8 •* 4 5 1 MIN ,9 4 4 9 C 4 5 2 MIN .9 4 5 0 8 4 5 3

A137

HINT.# 451 c 454MINT,# a k a s 457MinT.# 457 5 460H’INt .# 46 0 = 463MINT.# 463 * 466MINT,# 466 = 469MINT,# 469 = 472MINT.# 472 = 475MINT.# 475 = 478MINT.# 478 = 481MINT.# 481 = 484MINT,# 484 s 487MINT.# 487 = 490MINT.# 490 b 493MINT,# 493 = 496MINT.# 496 = 499MINT.# 499 = 502MINT.# 502 = 505MINT.# 505 = 508MINT.# 508 b 511

M I N T , # 452 - 4 5 5MINT,# 455 s 458 MinT.# 458 =_ 461

"MINt *# 461 = 464MINT,# 464 s 467 MINT.# 467 B 470 MINT,# 470 s 473 MINT,# 473 s 476 MINT,# 476 = 479MINT,# 479 s 482 MINT,# 482 s 485 MINT,# 4B5 b 488 MINT,# 488 = 491MINT,# 491 = 494MINT,# 494 = 497MINT,# 497 =' 500"MINT,# 500 s 503 MINT.# 503 = 506MINT,# 506 s “5 0 9 “ MINT,#

MINT.# 453 B 456 MINT.# 456 ® 459mint*# #59 s 462 MINj,# 462 p 465 MINT.# 465 = 468MINT.# 468 s 471 MINT.# 471 = 474MINT.# 474 = 477MINT.# 477 = 480MINT.# 480 p 483 MINT.# 483 s 486 MINT.# 486 = 489MINT.# 489 p 492 MINf.# 492 p 495 MINT.# 495 p 498 MINT . # 498 s ' " 501MINT.# 5oI p 5o4 MINT.# 504 p 507 MINt .# 507 p 510

yHE C|JBE COVERING i_1ST j

tHE 1-tH C mBE COVERING IS :8-CUBE B ( 256 * 511 > - 1

THIS CUr E COVERS THESE m INTERm S I256 511 257 258 259 260 g6i 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276.277 278 279 280 .281 282 ,283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 3o2303 304 305 306 307 308 309 310 311 312 313 314 315 316 3i7 3l8319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350351 352 353 354 355 356 357 358 359 360 36l 362 363 364 365 366

__ _ 367 368 .369 37£_37l_372 373 374 375 376 377 378 379 38q 381 382383 384 385 386 387 308 389 390 391 392 393 394 395 396 397 398399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 4i44l5 416 4l7 4l8 419 420 421 422 423 424 425 426 4 2 7 428 429 430431 432 433 434 435 436 ‘437 438 439 440 '44i 442 443 444 445 446447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462463 464 465 466 467 468 469 470 471 472 4.73 474_475„476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510

A138

T HE 2 - T H C U B E C O V E R I N G IS l8 *■

THIS CUBE

■c u b e ei COVERS

( 8 . 511 ) s THESE MINTERMS 1

8 511 9 10 11 12 13 14 15 24 25 26 27 28 29 3031 40 41 42 43 44 45 46 47 56 57 58 59 60 61 6263 72 73 74 75 76 77 78 79 88 89 90 91 92 93 9495 104 105 106 107 108 109 110 111 120 121 122 123 124 125 126

127 136 137 138 139 140 141 142 143 152 153 154 155 156 157 158159 168 169 170 171 172 173 174 175 184 185 186 187 188 189 190191 200 201 202 203 204 205 206 207 216 217 218 219 220 221 222223 232 233 234 235 236 237 238 239 248 249 250 251 252 253 254255 264 265 266 267 268"269 270 271" 280 "2Bl 282 283 “284 285 286287 296 297 298 299 300 301 302 303 312 313 314 315 316 317 318319 328 329 330 331 332 333 334 335 344 345 346 347 348 349 350351 360 361 362 363 364”365 366 367 376 377 378 379 300 381 382383 392 393 394 395 396 397 398 399 408 409 410 411 412 413 414415 424 425 426 427 428 429 430 431 440 441 442 443 444 445 446447 4 5 6 457 4 5 Q "459 460 461 462 463 472 "473“474 475 476 477 478479 488 489 490 491 492 493 494 495 504 505 506 507 508 509 510

THE " 3-TH CUBE COVERI NG is •8- c u b e = ( 2! . 511 ) = 1-

THIS CUBE1 COVERS THESE M INTERMS i -- - — ---- .- •2 5111 3 6 7 10 11 14 15 18 19 22 23 26 27 30

31 34 35 38 39 <42 43 46 47 50 51 54 55 58 59 6263 66 ' 67 ' 70 7f 74 ”75 78 7 9" 82 83' 86 87 90 91 9495 98 99 102 103 106 107 110 111 114 115 118 119 122 123 126

127 130 131 134 135 138 139 142 143 146 147 150 151 154 155 158159 162 163 166 167 170 171 174 175 178 179 182 183 186 187 190191 194 195 198 199 202 203 206 207 210 211 214 215 218 219 222223 226 227 230 231 234 235 238 239 242 243 246 247 250 251 254255 258 259 262 263 266 267 270 271 274 275 278 279 282 283 286287 290 291 294 295 298 299 302 303 306 307 310 311 314 315 318319 322 323 326 327 330 331 334 335 338 339 342 343 346 347 350351 354 355 358 359 362 363 366 367 370"“371" 374 375 378 379 382383 386 387 390 391 394 395 398 399 402 403 406 407 410 411 414415 4i8 419 422 423 426 427 430 431 434 435 438 439 442 443 446447 450 451 454 455 458 459 462 463 466 467 470 471 474 475 470

—479 482 483 486 487 490 491 494 495 498 499 502 503 506 507 510

THE 4-TH CUBE 'CO VER 11n g 'Ts J ---- ----- ----8 ■c u b e s { 4 » !511 > =

A139

THIS CUBE COVERS THESE MINTERMS I4 cil 5 6 7 12 13 14 Is 20 21 22 23 2fl 29 30

31 36 37 38 39 44 45 46 47 52 53 54 55 60 61 6263 68 69 70 71 76 77 78 79 84 " 85 86 87 92 93 9495 100 101 102 103 108 109 110 111 116 117 118 119 124 125 126

127 132 133 134 135 140 l4i 142 143 148 149 150 151 156 157 158~ “ 159 164 165 166 167 172 173 174 175 180 181 182 i'83 188 189 190

191 196 197 198 199 204 205 206 207 212 213 214 215 220 221 222223 228 229 230 231 236 237 230 239 244 245 246_?47 252^253 254 255 26^ 261^262 263 268“ 269“ 270 271“ 276 277 278 279 284 285 286 287 292 293 294 295 300 301 302 303 308 309 310 311 316 317 3l8319 324 325 326 327 332 333 334 335 340 341 342 343 348 349 350351 356 357 358 359 364 365 366 367 372 373 374 375 380 381 382383 388 389 390 391 396 397 390 399 404 405 406 407 412 4l3 4l4415 420 421 422 423 428429^ 430 431 436 437 438 439 444 445 446

447 452 453 454 455 460 461 462 463 468 469 470 471 476 477 478479 484 485 486 487 492 493 494 495 500 501 502 503 508 509 510

THE 5-TH CUBE COVERING IS :8-CU8E = ( 1 # 511 > = 1

THIS CUBE COVERS THESE MINTERMS I1 511 3 5 7 9 11 13 15 17 19 21 23 25 27 2931 33 35 37 39 41 43 45 47 49 51 53 55 57 59 6163 65 67 69 71 ,73 75 77 79 81 83 85 “ 07 89 91 9395 97 99 101 103 105 107 109 111 113 115 117 119 121 123 125

127 129 131 133 135 i37 l39 141 143 145 147 149 l51 153 155 157159 161 163 165 167 169 l7l 173 175 177 179 181 183 105 187 189191 193 195 197 199 201 203 205 207 209 211 213 215 217 219 221223 225 227 229 231 233 235 237 239 241 243 245 247 249 251 253255 257 259 26r 263 <265” 267 269 27l 273 275 277 279 201 283 205 287 289 291 293 295 297 299 301 303 305 307 309 311 313 315 317319 321 323 325 327 329 331 333 335 337 339 341 343 345 347 349

"351” 353 355 357 ”359 361 363 365 367 ”369 371 373 375 377 379 381383 385 387 389 391 393 395 397 399 401 403 405 407 409 411 4l3415 417 419 421 423 425 427 429 431 433 435 437 439 441 443 445

- 4 4 7 -4 49 451” 453 ”“"455 "457 459 461” 463”465 4 67" 469 471 “473 475 477479 481 483 485 487 489 49l 493 495 497 499 501 503 505 507 509

THE FINAL CHECKINGi25 105 iO9 3°9 309 309 409 3£>9 3o9

-____________ 22* 5S9 409 _4Q 9 _ 5 0 9 209 209 309 209 309 309 409209 309 309 ” 409 309 4Q 9 ™ 4 0 9 509— 209 “209 ““ 309 “ 209 309 309 409 209 309 309 409 309 409 409 509 109

209 209 309 209 309 309 409 209 309 309 409 3092S? IS9 ~509 209 209 309 209 “ 309 309 409 209 309_309 409 309 409 409 509 109 209 209 309 209 30930? 40? 20? 30? 30? 40? 30? 4?? 40? 50? i S ? 20?

A140

2 0 9 3 0 9 2 0 9 3 0 9 3 0 9 4 0 9 2 0 9 3 0 9 3 0 9 4 0 9 3 0 9 4 0 94 0 9 5 0 9 1 0 9 2 0 9 2 0 9 3 0 9 2 0 9 3 0 9 3 0 9 4 0 9 2 0 9 3 0 93 0 9 4 0 9 3 0 9 4 0 9 4 0 9 5 0 9 2 0 9 2 0 9 3 0 9 2 0 9 3 0 9 3 0 94 0 9 2 0 9 3 0 9 3 0 9 4 0 9 3 0 9 4 0 9 4 0 9 5 0 9 1 0 9 2 0 9 2 0 93 0 9 2 0 9 3 0 9 3 0 9 4 0 9 2 0 9 3 0 9 3 0 9 4 0 9 3 0 9 4 0 9 4 0 95 0 9 . 1 0 9 2 0 9 2 0 9 . 3 0 9 2 0 9 3 0 9 3Q9 409- . 2 0 9 3 0 9 3 0 94 0 9 3 0 9 4 0 9 4 0 9 5 0 9 1 0 9 2 0 9 2 0 9 3 0 9 2 0 9 3 0 9 3 0 94 0 9 2 0 9 3 0 9 3 0 9 4 0 9 3 0 9 4 0 9 4 0 9 5 0 9 1 0 9 2 0 9 2 0 93 0 9 2 0 9 3 0 9 3 0 9 4 0 9 2 0 9 3 0 9 3 0 9 4 0 9 3 0 9 4 0 9 4 0 95 0 9 1 0 9 2 0 9 2 0 9 3 0 9 2 0 9 3 0 9 3 0 9 4 0 9 2 0 9 3 0 9 3 0 94 0 9 3 0 9 4 0 9 4 0 9 5 0 9 1 0 9 2 0 9 2 0 9 3 0 9 2 0 9 3 0 9 3 0 94 0 9 2 0 9 3 0 9 3 0 9 4 0 9 3 0 9 4 0 9 . 5 0 9 109... . . 2 0 9 2 0 93 0 9 2 0 9 3 0 9 3 0 9 4 0 9 2 0 9 3 0 9 3 0 9 4 0 9 3 0 9 4 0 9 4 0 95 0 9 3 0 9 3 0 9 4 0 9 3 0 9 4 0 9 4 0 9 5 0 9 3 0 9 4 0 9 4 0 9 5 0 94 0 9 5 0 9 5 0 9 6 0 9 2 0 9 3 0 9 3 0 9 4 0 9 3 0 9 4 0 9 4 0 9 5 0 93 0 9 4 0 9 4 0 9 5 0 9 4 0 9 " 5 0 9 5 0 9 6 0 9 2 0 9 3 0 9 3 0 9 4 0 93 0 9 4 0 9 4 0 9 5 0 9 3 0 9 4 0 9 4 0 9 5 0 9 4 0 9 5 0 9 5 0 9 6 0 92 0 9 3 0 9 3 0 9 4 0 9 3 0 9 4 0 9 4 0 9 5 0 9 3 0 9 _ 4 0 9 4 0 9 5 0 94 0 9 5 0 9 5 0 9 6 0 9 2 0 9 3 0 9 3 0 9 4 0 9 3 0 9 4 0 9 4 0 9 5 0 93 0 9 4 0 9 4 0 9 5 0 9 4 0 9 5 0 9 5 0 9 6 0 9 2 0 9 3 0 9 3 0 9 4 0 93 0 9 4 0 9 4 0 9 5 0 9 3 0 9 4 0 9 4 0 9 5 0 9 4 0 9 5 0 9 5 0 9 6 0 92 0 9 3 0 9 3 0 9 4 0 9 3 0 9 4 0 9 4 0 9 5 0 9 3 0 9 4 0 9 4 0 9 5 0 94 0 9 5 0 9 5 0 9 6 0 9 2 0 9 3 0 9 3 0 9 4 0 9 3 0 9 4 0 9 4 0 9 5 0 93 0 9 4 0 9 4 0 9 5 0 9 4 0 9 5 0 9 5 0 9 6 0 9 2 0 9 3 0 9 3 0 9 4 0 93 0 9 4 0 9 4 0 9 ' 5 0 9 3 0 9 ‘ 4 0 9 4 0 9 “' “5 0 9 4 0 9 5 0 9 5 0 9 6 0 92 0 9 3 0 9 3 0 9 4 0 9 3 0 9 4 0 9 4 0 9 5 0 9 3 0 9 4 0 9 4 0 9 5 0 94 0 9 5 0 9 5 0 9 6 0 9 2 0 9 3 0 9 3 0 9 4 0 9 3 0 9 4 0 9 4 0 9 5 0 93 0 9 4 0 9 4 0 9 5 0 9 4 0 9 5 0 9 5 0 9 6 0 9 2 0 9 3 0 9 3 0 9 4 0 93 0 9 4 0 9 4 0 9 5 0 9 3 0 9 4 0 9 4 0 9 5 0 9 4 0 9 5 0 9 5 0 9 6 0 92 0 9 3 0 9 3 0 9 4 0 9 3 0 9 4 0 9 4 0 9 5 0 9 3 0 9 4 0 9 4 0 9 5 0 94 0 9 5 0 9 5 0 9 6 0 9 “ 2 0 9 3 0 9 " 3 0 9 “ 4 0 9 3 0 9 4 0 9 4 0 9 5 0 93 0 9 4 0 9 4 0 9 5 0 9 4 0 9 5 0 9 5 0 9 6 0 9 2 0 9 3 0 9 3 0 9 4 0 93 0 9 4 0 9 4 0 9 5 0 9 3 0 9 4 0 9 4 0 9 5 0 9 4 0 9 5 0 9 5 0 9 6 0 92 0 9 3 0 9 3 0 9 4 0 9 3 0 9 4 0 9 4 0 9 5 0 9 3 0 9 4 0 9 4 0 9 5 0 94 0 9 5 0 9 5*09 6 0 9

A141

i_i

XIMUK CUBE DIMENSION = 8

1 0 0 0 0 0 0 0 2 1 00 0 3 3 1 0 0 0 0 0 40 0 0 0 5 10 10 " 5 1 0 0

20 15 6 1 0 0 7 21 35 35 218 28 56 70 i56 .28 — 8 — - --- ---- ----

E c h a r a c t e r i s t i c s t8 5 8 5 8 5 8 5 8 5 5 1 8 5 6 5 88 5 8 5 8 5 8 5 8 5 8 5 8 1 8 1 88 5 8 5 8 1 8 5 8 5 B 5 8 5 6 5 86 1 8 5 ....e _1 8 5 8 5 8 5 8 i_ 6 88 5 8 5 8 5 7 1 8 5 8 5 8 • 5 B 5 88 5 8 5 8 5 8 5 8 5 8 5 8 5 8 5 88 1 8 5 8 5 8 5 8 1 8 5 8 5 0 5 88 5 7 1 8 5 8 5 8 5 8 5 8 5 8 5 88 5 8 5 8 5 8 5 6 5 8 5 7 1 8 5 88 5 8 5 8 5 B 5 8 5 8 5 8 5 6 5 88 5 B 5 8 5 8 5 8 5 B 5 8 5 8 5 88 5 8 5 8 5 6 5 0 5 8 1 8 1 8 5 88 5 8 1 8 5 B 5 8 5 8 5 8 5 6 5 68 5 8 5 8 5 8 5 ' 8 5 8 5 8 5 8 5 88 5 8 5 8 5 7 1 8 5 8 5 8 5 8 5 88 5 6 5 8 5 8 5 8 5 6 5 8 5 8 3 88 5 8 5 8 5 8 5 8 5 8 '5 " 8 5 8 5 88 5 8 5 7 1 8 5 8 5 8 5 8 5 6 5 88 5 8 5 8 5 5 8 5 8 5 8 5 8 5 88 i 8 5 8 5 .8 5 8 5 8 5 8 5 ~ B 5 8a 5 8 5 8 5 8 5 8 5 8 5 8 5 B 5 88 5 8 5 8 5 8 5 8 5 8 5 8 5 8 5 8

' 0 5 8 5 8 5 ~ 8 5 8 - 5 8 5 8 5 8 5 86 5 8 5 8 5 8 5 8 5 8 5 8 5 8 5 88 5 8 5 8 5 . 8 5 8 1 8 5 8 5 8 3 88 5 8 5 8 5 8 5 8 5 8 5 8 5 8 5 88 5 8 5 8 5 B 5 8 5 8 5 8 5 8 5 88 5 8 5 8 5 8 5 8 5 8 5 8 5 8 5 8

... 8 5 8 5 "S' 5 “ 8 ‘ 5 _ 8 - 5 “ 8 - 5- 0 - 5 8 5 88 5 8 5 8 5 8 5 8 5 8 5 8 5 8 5 86 5 6 5 8 5 8 ,5 8 3 8 5 8 5 6 5 88 5 8 5 8 5 8 --5 - 8 5 '8_ 5 __ 8 5 8 5 88 5 8 5 8 5 8 5 8 5 6 5 8 5 8 5 88 5 8 5 8 5 8 5 8 5 6 5 8 5 8 5 8B „ 5__ 8 -5 - 8 - 5 B 5 8 5 8 5 8 1 8 5 8 ~8 5 8 5 8 5 8 5 8 3 8 5 8 5 8 5 6a 5 8 5 8 5 8 5 8 5 8 5 8 5 8 5 88 5 8 5 8 5 8 -5 8 5 8 5 ~ 8 5 8 5 88 5 8 5 8 5 8 5 8 5 8 5 8 5 ’8 5 88 5 8 5 8 5 B 5 8 3 8 5 8 3 8 5 8

' 8 - 5 8 ~ 5 ~B 5 8 - 5 B 3 B 5~~ 8 5 8 5 ' 88 5 8 5 8 5 B 5 8 3 8 5 8 5 8 5 6

0607

55555155555155555555555555551555555~5555555

88B888BB888878 6 8 8 8 8 8 B 8 8 8 8 6 8 8 8 8 8 0 8 8 8 8 8 8 8 8 8 0

0461

5155515555551555555555555i5555555555555555

01

15n

888888888888880888888888888888888888888888

551555555555555555555555555555555555555555

A142

6 5 B 5 8 5 8 5 8 5B 5 5 8 5 8 5 8 38 5 6 5 8 5 8 5 8 Se 5 8 5 8 5 8 5 8 3B ”5 8 5

8 5 8 5 8 5 B 5 6 5 8 58 5 8 5 8 5 8 5 8 5 8 58 5 8 5 8 5 8 5 8 5 B 58 5 8 5 8 5 8 5 8 5 8 5

ALL CORE CUBES HAVE BEEN SEARCHED,,.AND THE MAXIMUM CUBE DIMENSION IS I '8

r.UBE ( 2 5 6 i. . 5 1 1 HAS..BEEN.FOUND,_____________ ___________ _CHECKING POINTS

6 4 6 4 6 4 6 4 9 9 9 9 9 9 9 9 , 9 9 9 9 9 9 9 9 9 9 9 9 . 9 9 9 9 . 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

_ _ 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

* 9 9 9 9 9 9 9 9 9 9 9 , 9 . 9 9„99 , 9 9 9 9 9 9 9 9 9 9 9 9 __99 9 9 . 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 _ 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 ,9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 “ 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 “ 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 .9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

A143

9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99999999 9999 9999 9999

ACCUMULATED AWAYNESS ••2 2 2 2 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0

..."... 0 ■" 0 0 ~ 0 __ 0_ 0 0 0 6 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 " b 0 " 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 o o 0 0 0 0 6 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 o o 0 0 0 0 0 0 0 00 __ 0 0

_ 0 ■'. 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 “ "'0 ' ■ 0 0 ..... ___ 0 0 0 0 0 ..... 0 00 0 0 0 0 0 0 0 0 0 0 0

f 0 0 0* 0 0 0 0 0 0 D 0 n0 0 O' 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 o o 0 0 0 0 b 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 D 0 0D 0 0 0 0 0 0 0 b 0 0 00 0 o o 0 0 0 0 0 c 0 00 0 0 0 0 0 0 0 0 0 0 00 0 o o 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 b b 0 00 0 0 0 0 0 0 0 b 0 0 00 0 0 0 0 0 0 0 0- 0 0 ...... 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 b 0 0 0

....... 0 __ 0 0 0 0 0 ... ;0 0 0 0 00 0 0 0 0 0 0 0 ' b 0 0 00 0 0 0 0 0 0 0 0. A 0 00 0 0 0

A144

CUBE < 2 5 6 , 5 1 1 ) IS SELECTED.

CUBE < 8 . 5 H ) HAS BEEN FOUND,. c h e c k i n g POINTS

1 2 8 1 2 8 1 2 8 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99999 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99999 9999 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9999 9 9 9 9 9999 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

~ ~ 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9999 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 ’ 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 * 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 ' 9 9 9 9 9 9 9 9 “ 9 9 9 9 “ 9 9 9 9 ” 9 9 9 9 9 9 9 ? 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 “ 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 ' 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 ' 9 9 9 9 9 9 9 9 “ 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 ? ~ 9 9 “9 9 9 9 9 9 “ 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 _ 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 “ 9 9 9 9 9 9 9 9 9 9 0 9 9 9 9 9 9 9 9 9 7 9 9 9 9 9 9 0 9 9 0 0 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

“9 9 9 9 9 9 9 9 9 9 9 9 ' 9 9 9 9 9 9 9 9 ’ 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 5 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 0 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

ACCUMULATED AWAYNESS I

A145

4 4 4 2 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 .... 0 0 0 . 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 d 0 0 00 0 0 0 0 0 0 0 0 0 0 0Q 0 0 0 0 0 0 0 0 ... 0 . 0 00 0 0 0 0 0 0 • 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 s 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 ..... 0 0 o ' 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 n 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0d 0 0 0 0 0 ' 0 0 0 0 0 p0 0 0 0 0 0 0 0 0 0 0 00 0 o* 0 0 0 0 0 0 0 0 00 0 t> 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 o 0 0 d b b 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 b 0 0 00 0 0 0 0 0 " 0 0 ~ 0 0 0 00 0 0 0 0 0 0 0 0 b 0 00 0 0 0 0 0 0 0 13. 0 0 00 o ' 0 0 0 0 0' 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 o ' 0 0 ■ ~ 0 0 0 0 0 '...0..... 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0

CUBE < 8 , 511 > IS SELECTED,CUBE < 2 T ' 5 U ~ > HAS BEEN''FOUND, "

CHECKING P OI NTS _______ _ . . _ . — - -192 9999 192 9999 9999 9999 9999 9999 9999 9999 9999 9999

A146

9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 ' 9 9 9 9 9 9 9 9 9 9 9 9 ~ 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 _99 9 9 _9999_ _9_9 9 _9__9_9_9 9_ _9_9_99_ _?_9 9 9 _ 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

" ' 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 ’ 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 ? ? _ ? 9 9 9 _ 9 ? 9 ^ J ? 9 9 9 9 9 9 9 _ 9 9 9 9 _ 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 ' 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 ? 9 9 9 9 9 9 9 9 9 9 9 9 9 ? 9 ? 9 9 9 ? 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9999 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 ^ 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 ’ 9 9 9 9 9 9 9 9 ' 9 9 9 9 9 9 9 9 ' 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 " 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 ?9 9 9 9 9 9 9 9 9 9 9 ? 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 ? 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9999 9 9 9 9

ACCUMULATED AWAYNESS »__________________________________________________________

6 4 6 2 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

— O' ' 0 O' 0 " 0 ~ 0 0 ■ 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

A147

0 0 0 0 0 0 0 0 0 0 o 00 o 0 0 0 0 0 0 0 0 o 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0o o o o o o o o o o o o 0 0 0 0 0 0 0 0 0 0 0 0 o 0 0 0 0 0 0 0 0 0 0 0o o o o o o o o o o o o0 0 0 o o _ _ o _____0 0 0 0 0 0

■ o o o o o o o o o b o oo o o o o o o o o o o oo o o o o o o o o o o oo ----- 0 0 0----0 0 0 0 0 0 0 0o o o o o o o o o o o o o o o o o o o o o o o o0 0 o o o 0... 0 0 f 0 0 0o o o o o o o o o o o o o o o o o o o o b o o o o o o o o o o o b b o o o o o o o o o o o o o o0 0 0 0 0 0 0 ____ o _ ____ o o o o

- o 0 0 0 o O ' 0 0 o o o bo o o o o o o o o o o o o _Jp o 0 0 0 _ 0 _0 b . O 0. 00 0 0 “o' o o o o o o o oO O O O O O O O O O O Oo o o o o o o o o o o o o o o o o o o o o o o o

# 0 0 0 0 0 0 0 0 0 0 0 00 _0 o’____ 1 _____0 0 0_______ 0____ 0 0 0 0o o o o o o o o d o o oo o o o o o o o o o o o

_ o 0 ___ 0 _____D_____CL_____0____0 _ _____Q____ 0_____ 0____o ____ 0O O O O O O O O O O O Oo o o o o o o o o o o oo 0 0 _ 0 0 0 _ 0______ 0 0 0 0 0o o o ~b" o o o o S o o o0 0 0 0

CyBE ( 2 ; 5 1 1 ) I S S E LE C t E D ,

CUBE ( 4 . 5 1 1 )......... h a s BEEN FOUND. ........................

CH.ECKJ NG _P ° 2 ^ 6 S 9 9 9 9 _ 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 _ ? 9 ? 9 _ 9 9 9 9 _ 9 ? 9 ? 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 ? 9 9 99 9 9 9 9 9 9 9 " 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9999 '9999 9 9 9 ? 9 9 9 9 9 9 9 ? 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 ? 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 ? ; ? 9 9 9 9 ? 9 9 j 9 9 9 9 9 9 9 9 9 9 9 ? 9 9 9 ? 9 9 9 9 9 9 ? P YVV*

1

A148

9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 ^ 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 ? 9 9 9 9 9 9 9 ? 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 ? 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

-^9999 9 9 9 9 ? ? 9 ? _ ? 9 9 9 _ 9 9 9 9 _ ? 9 ? 9 _ 9 ? 9 9 _ 9 ? 9 9 _ 9 ? 9 ? J ? ? ? _ 9 ? ? ? 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 . 9 9 ? 9 „ .9 9 9 9 . 9 9 9 9 - 9 9 9 9 . . 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 . 9 9 9 ? 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9999 9 9 9 ? 9 9 9 9 ? 9 ? 9 „ ? 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 ' 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 ? 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 ? 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 ? 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 . 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 ? 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 ? 9 9 9 9 9 9 9 9 9 9 9 9 9 9 ? 9 9 9 9 9 9 9 ? 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

I 9 9 9 9 9 9 9 9 99S9 99 9 9

ACCUMULATED AWAYNESS :6 4 6 2 0 0 0 0 0 0 0 0 o o o o o o o o o o o o o o o o o o o o o o o o0 0___0 __ 0____o______0___ 0 0____0 0 0 00 0 0 0 0 0 0 0 0 0 0 0O O O O O O O O O O O OO O O O O O O O O O O O0 0 0 0 " 0 0 0 0 b 0 0 0O O O O O O O O O O O O0 _ o _0 _ 0_________0____ 0___ jl____0 _____0_____0___ 0 00 0 0 0 0 0 0 0 0 0 0 0O O O O O O O O O O O O O O O O O O O O O O O O0 0 0 0 0 0 0 0 b 0 0 0O O O O O O O O O O O O0 0 0 0 0 0 0 0 ____ 0____ 0 0 00 0 0 0 0 0 0 0 0 0 0 0O O O O O O O O O O O OO O O O O O O O O O O O

A149

CUBE (

CUBE I

CHECKING

0 0 0 0 0 0 p 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 fl0 0 0 0 0 0 ~ 0 0 b 0 V

00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 b 0 00 0 0 0 0 0 0 0 b 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 ' " 0 0 0 0 0 00 0 0 0 0 0 0 0 b 0 00 0 0 0 0 0 • 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 b 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 O' 0 0 ' 0 0 00 0 0 0 0 0 0 0 b 0 00 0 0 0 0 0 0 0 b 0 00 0 0 0 0 0 0 ■ o~ 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 O' 0 0 0 b 00 0 y 0 0 0 0 0 0 0 0 00 0 ’ 0 0 0 0 0 0 0 0 00 0 0 0

« 511 > IS SELECTED,

1 , gll ') HAS BEEN FOUND,

p o i n t s ............9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99?9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99?

~ 9999~“9999 9999 9999 9999~9999~9999 9999 9999 9999 9999 99? 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 99?9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 999

- 9 9 9 9 - 9 9 9 9 9 9 9 9 9999 9999 ’9999 9999 9999 9999 9999 9999 99?9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999999 9999 9999 9999 9999 9999 9999j 9999 9999 9999 9999 9999999 9999 9999“9999” 9999 99 9 9 9999 9999 9999 9999 9999 9999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999999 9999 9999 9999"9999~9999'9999 9999 9999 9999 9999 999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 999

A150' ■iiJi

9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 oqcq *9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99oo o!9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 oo o9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 o,9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 o<

____________ _ ? 9 9 9 _ 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 Q 9 9 *9 9 9 9 9 9 9 9 9 9 9 9 " 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 ? 9 9 9 9 9 9 9 9 9<9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 919 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9<9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9'9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9'9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9'9 9 9 9 9 9 9 9 " 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9' 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9'9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9'9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9'9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9<9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 ^ 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9< 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 ' 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9<9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9<9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9<9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9< 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9<9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9<9999 9999 9999 9999 9999 9 9 9 9 9 9 9 9 9999 9 9 9 9 9 9 9 9 9 9 9 9 9<9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9999 9<9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9<9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9<

I 9 9 9 9 9 9 9 9 9*999 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 959 9 9 9 9 9 9 9 9 ’9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 959 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 ?9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 959 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 959999 9999 9999 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 999 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99

________________9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 _________________________________________________

a c c u m u l a t e d AWAYNESS s

0 0 0 0 0 00 0 00 0 00 0 0 0 0 00 0 00 0 0 0 0 0

o o o o o o o b o0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0

8 4 6 2 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

A151

0 0 0 0 0 " 0 0 0 b 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 .... 0 0 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 ■ 0 "V~~ 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 " ~ 0 Q 0 0 0 0 ' 0 ......... 0 0" " 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 • 0 0 0 0 0 00 - 0 0 ~ ” 0" 0 - 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0

CUBE ( l , 511 ) IS SELECTED,

I ?•♦FORTRAN *« STOP /LOGOFF

C E 4 2 0 LOGOFF AT 0 9 .11 ON .4 1 / 2 3 __F0.R_TSN_99.7_Z...

C E 4 2 1 CPU TIME USEDJ ' 0 0 0 6 3 8 . 5 3 5 5 SECONDS,