Odd Order Pandiagonal Latin and Magic Cubes in Three and Four Dimensions

Abstract

Explicit algorithms are developed for constructing odd order n pandiagonal latin cubes in 3 and 4 dimensions, and these are used to construct pandiagonal magic cubes and 4 dimensional hypercubes, respectively. It is established that these structures exist in 3 dimensions only for odd orders n not less than 11, and in 4 dimensions only for odd orders n not less than 17. Estimates for the number of such structures as functions of n in 3 and 4 dimensions are also presented.