A Family of \(N \times N\) 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 \(n \times n\) Tuscan-\(2\) squares exist for infinitely many composite values of \(n+1\). It is shown here that if \(p\) is a prime and \(p \equiv 7 \pmod{12}\) or \(p \equiv 5 \pmod{24}\), then Tuscan-\(2\) squares of side \(2p\) exist. If \(p \equiv 7 \pmod{12}\), clearly \(2p + 1\) is always composite and if \(p \equiv 5 \pmod{24}\), \(2p+1\) is composite infinitely often. The squares constructed are in fact Latin squares that have the Tuscan-\(2\) property in both dimensions.