Correction: SYMMETRY IN AMERICAN FOLK ARTSource: The Arithmetic Teacher, Vol. 38, No. 4 (DECEMBER 1990), p. 17Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/80000199 .

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Six hard pieces revisited

In the October 1989 issue (pp. 34-35) Bill Jamski investigates the interesting challenge of building a square with one to seven tan- gram pieces. He outlines the possible solu- tions for each quantity except six. His table 1 suggests an unproved hypothesis that L (little) triangles of powers of 2 (i.e., 2, 4, 8, or 16) are necessary to form a square. Jamski then shows that with six of the tangram pieces one can form only twelve, fourteen, or fifteen L triangles, and hence a six-piece square is not possible. His challenge to find a formal proof of his powers-of-2 conjecture prompted me to tackle the problem in a slightly different way.

Figure 1 lists and labels the pieces as Jam- ski did, but I have added the lengths, in terms of a and b, of the sides of the various pieces. One can easily notice that the sides of each piece are multiples of a or b and that a < b, since a is the leg and b is the hypotenuse of the L triangle.

Let us now compare the five-piece square with the seven-piece square. Figure 2 pres- ents the solution Jamski gave for these two squares. The five-piece square has sides of length 2a, and the seven-piece square has sides of length 2b. The area of a six-piece square must be less than that of the seven- piece square. The sides of a six-piece square would have to satisfy the following two condi- tions:

■HHgyyyj a a

Five pieces / a / S a / p JL

XXL a /

XXL M '|* 2a

Seven pieces b b

V/V L 2b B Xi

^ / ' P

/ X b /

B X

2b

EB^^^SI

*[>k ^ S • a a a

a/^P% ~A b /'2a / v/ M X b L X b

2«-»V B ! x ■ x

2a

/ I ' 2a

1. The side is a combination of a and b. 2. The combination is greater than 2a and less

than 2b.

Since we are given that a < b, we can con- clude that

a + a < b + a and

a + b < b + b; hence,

2a < a + b < 2b.

The length of the sides of the six-piece square must be a + b because it is the only combination that satisfies both conditions. Such a square could be constructed only by putting a figure with a side a beside a figure with a side b; hence, we cannot use the B tri- angles that have sides of 2a and 2b. Since two such triangles are given, we must eliminate both pieces, and we can therefore conclude that the six-piece square is impossible.

Roger R. Carbone Manitoba Department of

Education Winnipeg, MB R3G 0T3

Jamski responds: Carbotte's solution to the problem is very interesting and differs from other solutions sent to me. It seems that mathematics educators are interested in prob- lem solving and good problems not only for their students but also for themselves. I take that interest as a sign that we have good peo- ple in the field.

Corrections

The following corrections should be noted in ''Symmetry in American Folk Art" (Septem- ber 1990): (a) in table 2 eight lines of symme- try should be designated for design A-4, the center flower; (b) the credit for figure 7 should have been "Students in the fourth-grade class of Iris Mitchell, PS 189M, New York, N.Y."; (c) the text near the bottom of page 10, in ref- erence to figure 3, should read, "The upper- right corner is different from the other three"; (d) Sam Zaslavsky should be credited with the photographs that appear in figures 2, 3, 6, 7, and 11; (e) Lawrence Hill Books, Brooklyn, N.Y., now publishes Claudia Zaslavsky' s Af- rica Counts: Number and Pattern in African Culture, listed in the Bibliography.

In NCTM journals

Readers of the Arithmetic Teacher might en- joy the following articles in this month's Mathematics Teacher: • "A Mathematics Magazine and Fair," Robert

R. Spieler • "Sharing Teaching Ideas: Mathematics - an

International Language," Philinda Stern Denson W

Statement of ownership, management and circulation (Required by 39 U.S.C. 3685). 1A. Title of publication, Arithmetic Teacher. IB. Publication no., 0004 136X. 2. Date of filing, 20 September 1990. 3. Frequency of issue, Monthly - September through May. 3A. No. of issues published annually, nine. 3B. Annual subscription price, $45. 4. Complete mailing address of known office of pub- lication, 1906 Association Drive, Reston, VA 22091- 1593, Fairfax County. 5. Complete mailing address of the headquarters of general business offices of the pub- lishers, same as #4. 6. Full names and complete maijing address of publisher, editor, and director of publica- tions. Publisher, National Council of Teachers of Math- ematics, 1906 Association Drive, Reston, VA 22091- 1593. Editor, none. Director of Publications, Harry B. Tunis, 1906 Association Drive, Reston, VA 22091-1593. 7. Owner, National Council of Teachers of Mathemat- ics, 1906 Association Drive, Reston, VA 22091-1593. 8. Known bondholders, mortgagees, and other security holders owning or holding 1 percent or more of total amount of bonds, mortgages or other securities, none. 9. The purpose, function, and nonprofit status of this or- ganization and the exempt status for Federal income tax purposes have not changed during preceding 12 months. 10. Extent and nature of circulation. Average no. copies each issue during preceding 12 months. A. Total no. copies, 50 813. Bl. Paid and/or requested circulation, sales through dealers and carriers, street vendors and counter sales, none. B2. Paid and/or requested circula- tion, mail subscription, 43 321. C. Total paid and/or re- quested circulation, 43 321. D. Free distribution by mail, carrier or other means, samples, complimentary, and other free copies, 1 250. E. Total distribution, 44 571. Fl. Copies not distributed, office use, left over, unac- counted, spoiled after printing, 6 242. F2. Copies not distributed, return from news agents, none. G. Total, 50 813. Actual no. copies of single issue published near- est to filing date. A. Total no. copies, 51 497. Bl. Paid and/or requested circulation, sales through dealers and carriers, street vendors and counter sales, none. B2. Paid and/or requested circulation, mail subscription, 42 564. C. Total paid and/or requested circulation, 42 564. D. Free distribution by mail, carrier or other means, samples, complimentary, and other free copies, 1 250. E. Total distribution, 43 814. Fl. Copies not dis- tributed, office use, left over, unaccounted, spoiled after printing, 7 683. F2. Copies not distributed, return from news agents, none. G. Total, 51 497. 1 1. 1 certify that the statements made by me above are correct and complete. James D. Gates, Business Manager.

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