Some Results on the Existence of Squares

In order to investigate the spectrum of skew Room squares, the authors obtain some preliminary results in preparation for a recursive attack. Some of these seem to be of interest in their own right. For example, if N(ν) denotes the maximum number of mutually orthogonal Latin squares of side ν, it is shown, by direct construction, that N(82) ⩾ 8 and N(100) ⩾ 8 and, by using recursive constructions, that N (ν)⩾ 8 for ν ⩾ 9445. It is also shown that for ν = 2 α t + 1, (2, t)= 1, aL α ≠ 1, 2, 6, 7, there exists a skew Room square of side ν.