Magic cubes and the 3-adic zeta function

Abstract

In this paper, we make some general observations about the construction of magic N-cubes. It then appears, in light of these observations, that the 3-adic zeta function can be used to construct an infinite class of magic cubes. As there is no need to rush into a mass of technical details when treating a subject that has always been regarded as fun, in the first two sections I will attempt to motivate the general discussion by describing two known methods of constructing magic N-cubes. In doing so, I will enlarge upon methods appearing in Kraitchik's book [3], in recreational paperbacks I read as a child, including [4], and in the article [1]. In the third section, I will show how to combine these methods with another technique from I1] to obtain a more general perspective of the problem of constructing magic N-cubes. Finally, in the last section I will illustrate this more general point of view using the 3-adic zeta function. The results of this article were obtained in 1984 while preparing a colloquium lecture on magic N-cubes at SUNY Stony Brook. I would like to thank the mathematics department at Stony Brook for its hospitality during the period 1983-1986. I would also like to thank Bob Messer and other correspondents from the electronic mailing list TEXhax for their help in typesetting the magic squares and Denys Duchier for his help typesetting the tessaract in Figure 10. I would also like to thank Larry Washington for carefully checking the proof of the main result of this paper and pointing out an error. Finally, I would like to thank Roger Howe and Walter Feit for their help in making the facilities of Yale University available to me while I typeset this article.