The Spectrum of Room Cubes

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 in the array. If t;;;.: 2 is an integer, a Room t-cube of side n is at-dimensional array, each cell of which is either empty or contains an unordered pair of two distinct elements chosen from a set of size n +1, such that each 2-dimensional projection is a Room square of side n. In this paper we consider Room 3-cubes which we refer to as Room cubes. Let R3 ={n: there exists a Room cube of side n}. Room t-cubes can be equivalently described as certain types of Latin squares. Such a formulation has the advantage of being easier to visualize if t;;;.: 3. A Latin square L of side n is an n by n array of cells, each of which contains exactly one symbol chosen from a set S of size n, such that each element of S occurs once in each row and once in each column of L. Suppose the rows and columns of L are indexed by the members of S. L is said to be idempotent if L(s, s) = s for each s E S. L is said to be symmetric if L(s, s') =L(s', s) for every {s, s'}£ S. Now impose any linear ordering on S. Two Latin squares L and M, on the symbol set S, are said to be orthogonal symmetric Latin squares provided Land M are both symmetric and idempotent, and for every (s, s') E S X S there exists at most one cell (sh s2) with s1 < s2 such that (s, s') = (L(sh s2), M(sh s2)). We refer to a set of t orthogonal symmetric Latin squares as t pairwise orthogonal symmetric Latin squares (POSLS) if each pair of squares is orthogonal symmetric. Horton [9] establishes the following.