Magic cubes and the 3-adic zeta function (Q1200113)

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 \textit{M. Kraitchik}'s book [Traité des carrées magiques, Paris (1930)], in recreational paperbacks I read as a child, including \textit{J. S. Meyer} [Fun with mathematics, Cleveland (1952)], and in the article (*) \textit{A. Adler} and \textit{S.-Y. Robert Li} [Am. Math. Mon. 84, 618-627 (1977; Zbl 0389.05018)]. In the third section, I will show how to combine these methods with another technique from (*) 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.