\(d\)-dimensional hypercubes and the Euler and MacNeish conjectures (Q1346764)

Euler conjectured that if \(n\) is of the form \(n= 2(2k+ 1)\), then there is no pair of mutually orthogonal Latin squares (MOLS) of order \(n\). While this is known to be true for \(k= 0\) and 1, it is known to be false for all \(k\geq 2\). \textit{R. C. Bose}, \textit{E. T. Parker} and \textit{S. S. Shrikhande} disproved Euler's conjecture for \(n\geq 10\), see [Can. J. Math. 12, 189-203 (1960; Zbl 0093.319)] and Chapter 11 of the reviewer's joint book with \textit{A. D. Keedwell} [Latin squares and their applications (1974; Zbl 0283.05014)] devoted completely to the history of the disproof of Euler's conjecture. \textit{H. F. MacNeish} conjectured that if \(n= q_ 1q_ 2\cdots q_ h\) is the factorization of \(n\) into prime powers with \(q_ 1> q_ 2\cdots > q_ h\) then the maximum number of MOLS of order \(n\) is \(q_ 1- 1\), see [Ann. Math. 23, 221-227 (1923; JFM 49.0041.05)]. (Further details can be found in the joint book of the reviewer with A. D. Keedwell mentioned above.) The MacNeish conjecture is true only when \(n\) is a prime power or when \(n= 6\). In this paper the authors consider a \(d\geq 2\) dimensional generalization of the conjectures of Euler and MacNeish. The main result of the paper is the construction of mutually orthogonal \(d\geq 2\) dimensional hypercubes of order \(n\) from sets of MOLS of order \(n\). The authors were able to prove the falsity of both Euler's and MacNeish's conjecture for arbitrary dimension.