Calendar commentsAuthor(s): Douglas RogersSource: The Mathematics Teacher, Vol. 92, No. 4 (APRIL 1999), p. 351Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27970988 .

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Calendar comments Problem 6, May 1998 Problem 6 in the May 1998 "Calendar" states the following:

Balls numbered 1 through 6 are arranged in a difference triangle. Note that the absolute differences between successive balls appear below them. Arrange balls numbered 1 through 10 in a difference triangle.

This problem, under the name

"pool-ball triangles," has an

interesting history, going back in

published form at least to a column by Martin Gardner in Scientific American, which has been reprinted in Gardner's collection Penrose Tiles to Trapdoor Ciphers, a revised edition of which was issued by the Mathematical Association of America in 1997. The problem seems to have resurfaced in a French magazine for high school students around 1993 and was then brought to my attention by the late Germain Kreweras, resulting in the three research papers mentioned by Gardner on page 310 of his book.

The problem is related to a familiar problem about repeated differencing of integers in a cyclic arrangement, where the ends wrap around, sometimes called Ducei sequences. Some history of both problems appears in "Regular perfect systems of sets of iterated difference," by G. M. Hamilton, I. T. Roberts, and D. G. Rogers, which appeared in the European Journal of Combinatorics in 1998.

It is amusing?and encourag

ing?that a problem in the "Calendar" can be the subject also of lengthy articles in research journals; after all, everyone has to begin somewhere.

Problem 31, May 1997 Problem 31 in the May 1997 "Calendar" read as follows:

Find numbers A, B, C, D, E, and F for the six sectors so that the total number in a sector, or the total of the numbers in a set of adjacent sectors, gives all the integers from 1 to 27 inclusive.

Much as the previous problem was related to one in which the numbers were arranged circularly, so too this problem is a circular version of a well-known problem where the arrangement is linear, perhaps best known under the names of Golomb rulers or

difference sets. Brian Hayes gave an account of this linear version

in his regular column on computing science in the March-April 1998 issue of The American Scientist, and other references are given in the previously mentioned article by Hamilton, Roberts, and Rogers. Whereas problem 6 in the May 1998 "Calendar" seems

purely recreational, systems of difference sets find applications in coding theory, the layout of radio telescopes, and even missile guidance. However, a similar style of mathematical reasoning has been used for both.

Readers might like to try a

prototype problem discussed by D. W. Bange, A. E. Barkauskas, and Peter J. Slater in "Sequentially Additive Graphs" on pages 235-41 of volume 44 (1983) of Discrete Mathematics: label points on a circle with the numbers 1 through 2n in such a way that each number appears either labeling a point or as the sum of two adjacent labels.

Douglas Rogers drogers@cs. bgsu.edu Croxley Green United Kingdom, WD3 3HT

(Continued on page 352)

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