The non-existence of a pair of mutually orthogonal Latin squares of order six is a well-known result in the theory of combinatorial designs. It was conjectured by Euler in 1782 and was first proved by Tarry [4] in 1900 by means of an exhaustive enumeration of equivalence classes of Latin squares of order six. Various further proofs have since been given [1, 2, 3, 5], but these proofs generally ...

Let N(n) denote the maximum number of mutually orthogonal Latin squares of order n. It is shown that for large n,

A multi-latin square of order n and index k is an n×n array of multisets, each of cardinality k, such that each symbol from a fixed set of size n occurs k times in each row and k times in each column. A multi-latin square of index k is also referred to as a k-latin square. A 1-latin square is equivalent to a latin square, so a multi-latin square can be thought of as a generalization of a latin ...

A Room square of side n is an n by n array of cells, whose entries are chosen from a setS of n + 1 objects called symbols, which satisfies the following conditions: (i) every cell of the array is either empty or contains an unordered pair of distinct symbols from S; (ii) each symbol occurs in every row and in every column of the array; (iii) every unordered pair of symbols occurs precisely once...