# mathematical entertainments

of 3/3
Steven H. Weintraub* For the general philosophy of this section, see Volume 9, No. 1 (1987). A bullet (o) placed beside a problem indicates a submission without solution; a dagger (~) indicates that it is not new. Contrib- utors to this column who wish an acknowledgment of their contri- bution should enclose a self-addressed postcard. Contest entries and problem solutions should be received by I May 1989. The prize (a free subscription to the Intelligencer) for the best contribution to this column in 1988 is awarded to Krzysztof Cie- sielski for Problem 88-3, which appeared in Volume 10, No. 1. Such a prize will again be awarded for the best contribution in 1989. The winner of Contest 89-1 below will be awarded a free Springer book of his~her choice, up to a value of \$60. Problems Mathematicians' birth years: Contest 89-1 by the Column Editor Give the year of birth of each of the following mathe- maticians: 1. L. Euler 2. J.-L. Lagrange 3. C.-F. Gauss 4. A. L. Cauchy 5. P. L. Chebyshev 6. B. Riemann 7. S. Lie 8. H. Poincar4 9. D. Hilbert 10. G. H. Hardy 11. S. Lefschetz 12. J. E. Littlewood 13. S. A. Ramanujan 14. S. Banach 15. J. von Neumann 16. K. GOdel value given in an entry, then the score of the entry will be f(xl,yl) + 9 9 9 + f(x16,Y16). Of course, low score wins. (Ties will be broken by lot. Joint entries are wel- come, but readers are on their honor not to consult reference materials.) Repeated subtraction: Problem 89-2 by Henry Lulli (Gardena High School, USA) For a 4-tuple of non-negative integers A = (al,a2,a3,a4), let D(A) = (la2 - a,l,la3 - a2l,la4 - a3Hal - a4l). Given such a 4-tuple A, not identically zero, define a sequence {A~ by A0 = A, A i+l = D(Ai) for i I> 0. Let h = h(A) be the unique non-negative integer such that A h = (s,s,s,s) for some positive integer s = s(A). Determine all possible values for the pair (h,s). Contest entries will be scored as follows: Let fix,y) = - 5 if x = y; otherwise fix,y) = min (Ix - yl,30). If x i is the true year of birth of mathematician i, and Yi the * Column editor's address: Department of Mathematics, Louisiana State University, Baton Rouge LA 70803-4918 USA THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. I 9 1989Springer-Verlag New York 53

Post on 23-Aug-2016

215 views

Category:

## Documents

Embed Size (px)

TRANSCRIPT

• Steven H. Weintraub*

For the general philosophy of this section, see Volume 9, No. 1 (1987). A bullet (o) placed beside a problem indicates a submission without solution; a dagger (~) indicates that it is not new. Contrib- utors to this column who wish an acknowledgment of their contri- bution should enclose a self-addressed postcard. Contest entries and problem solutions should be received by I May 1989.

The prize (a free subscription to the Intelligencer) for the best contribution to this column in 1988 is awarded to Krzysztof Cie- sielski for Problem 88-3, which appeared in Volume 10, No. 1. Such a prize will again be awarded for the best contribution in 1989. The winner of Contest 89-1 below will be awarded a free Springer book of his~her choice, up to a value of \$60.

Problems

Mathematicians' birth years: Contest 89-1 by the Column Editor

Give the year of birth of each of the following mathe- maticians:

1. L. Euler 2. J.-L. Lagrange 3. C.-F. Gauss 4. A. L. Cauchy 5. P. L. Chebyshev 6. B. Riemann 7. S. Lie 8. H. Poincar4 9. D. Hilbert

10. G. H. Hardy 11. S. Lefschetz 12. J. E. Littlewood 13. S. A. Ramanujan 14. S. Banach 15. J. von Neumann 16. K. GOdel

value given in an entry, then the score of the entry will be f (x l ,y l ) + 9 9 9 + f(x16,Y16). Of course, low score wins. (Ties will be broken by lot. Joint entries are wel- come, but readers are on their honor not to consult reference materials.)

Repeated subtraction: Problem 89-2 b y Henry Lulli (Gardena High School, USA)

For a 4-tuple of non-negative integers A = (al,a2,a3,a4), let

D(A) = (la2 - a,l,la3 - a2l,la4 - a3Hal - a4l).

Given such a 4-tuple A, not identically zero, define a sequence {A~ by A0 = A, A i+l = D(Ai) for i I> 0.

Let h = h(A) be the unique non-negat ive integer such that A h = (s,s,s,s) for some positive integer s = s(A). Determine all possible values for the pair (h,s).

Contest entries will be scored as follows: Let f ix ,y) = - 5 if x = y; otherwise f ix ,y) = min (Ix - yl,30). If x i is the true year of birth of mathematician i, and Yi the

* C o l u m n editor's address: Depar tment of Mathematics, Louisiana State University, Baton Rouge LA 70803-4918 USA

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. I 9 1989 Springer-Verlag New York 53

• 54 THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1, 1989

• HUYGENS' PRINCIPLE AND HYPERBOLIC WAVE EQUATIONS Paul G/inther

For t he f i rs t t ime in b o o k form, the a u t h o r p r e s e n t s t he r e s u l t s o b t a i n e d b y severa l m a t h e m a t i - c i a n s t o w a r d s the so lu t i on of H a d a m a r d ' s p r o b l e m c o n c e r n i n g the va l id i ty of H u y g e n ' s p r inc ip le for hype rbo l i c d i f ferent ia l equa t ions .

1988, 847 pages, \$69.00 ISBN: 0-12-307330-8

ALGEBRAIC D-MODULES A. Borel e t a l .

1987, 355 pages, \$39.50 ISBN: 0-12 - 117740-8

BEILINSON'S CONJECTURES ON SPECIAL VALUES OF L-FUNCTIONS edited by M. R a p o p o r t , N. S c h a p p a c h e r , and P. Schne ider

This work p rov ides a t h o r o u g h i n t r o d u c t i o n to a c en t r a l topic in a r i t h m e t i c a lgeb ra i c geomet ry .

1988, 396 pages, \$37.50 ISBN: 0-12-581120-9

ARITHMETIC DUALITY THEOREMS J a m e s S. Milne

1986, 432 pages, \$39.95 ISBN: 0-12-498040-6

@ A C A D E M I C P R E S S

Harcourt Brace Jovanovich, Publishers

ANALYTIC PROPERTIES OF AUTOMORPHIC L-FUNCTIONS S t e p h e n Gelbart and Freydoon Shahidi

This vo lume p u t s in to p e r s p e c - t ive r e c e n t w o r k on the ana ly t i c a l p r o p e r t i e s of t he a u t o m o r p h i c L- func t ions a t t a c h e d to r educ t ive a lgeb ra i c g roups .

1988, 140 pages, \$18.75 ISBN: 0-12-279175-4

THE IWASAWA THEORY OF ELLIPTIC CURVES WITH COMPLEX MULTIPLICATION Ehud de Shalit

1987, 154 pages, \$19.50 ISBN: 0-12-210255-X

B o o k M a r k e t i n g D e p a r t m e n t #37019, 1250 S ix th A v e n u e , San Diego , C A 9 2 1 0 1 C a l l T o l l Free 1-800-311-5068 ~ subject to change without notice. CB/ES #37019.

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1, 1989 55