Graeco-Latin squares with embedded balanced superimpositions of Youden squares Original Research Article

Parkerʹs (1959) first example of a 10 × 10 Graeco-Latin square incorporates 4 balanced superimpositions of 3 × 7 Youden squares. Such superimpositions of size s × (2s + 1), where s is odd and (2s + 1) is prime, can be of two types, distinguished by the values taken by an invariant formed from the incidence matrices for the superimpositions. All four of the superimpositions in Parkerʹs example are of Type 1. We now give 10 × 10 Graeco-Latin squares similar to Parkerʹs, but with x (= 0, 2, 3) superimpositions of Type 1 and 4 − x of Type 2. Our 10 × 10 examples with x = 0, 2, 3 and 4 are shown to be special cases of constructions for Graeco-Latin squares of order 3s + 1; Graeco-Latin squares with x − 1 are shown to be impossible.