A FAMILY OF N X N TUSCAN-2 SQUARES WITH N + 1 COMPOSITE

Golomb and Taylor (joined later by Etzion) have modified the notion of a complete Latin square to that of a Tuscan-k square. A Tuscan-k square is a row Latin square with the further property that for any two symbols a and b of the square, and for each m from 1 to k, there is at most one row in which b is the m(th) symbol to the right of a. One question unresolved by a series of papers of the authors mentioned was whether or not n x n Tuscan-2 squares exist for infinitely many composite values of n + 1. It is shown here that if p is a prime and p = 7 (mod 12) or p = 5 (mod 24), then Tuscan-2 squares of side 2p exist. If p = 7 (mod 12), clearly 2p + 1 is always composite and if p = 5 (mod 24), 2p + 1 is composite infinitely often. The squares constructed are in fact Latin squares that have the Tuscan-2 property in both dimensions.