How many 7 × 7 latin squares can be partitioned into Youden squares? Original Research Article

Youden (1940) gave examples of Latin squares partitioned into Youden squares. However, the literature seems to have been silent on how many Latin squares of appropriate sizes can be partitioned in this way. We now show that, of the 564 isotopy classes (transformation sets) of 7 × 7 Latin squares, 29 consist of squares that can be partitioned into 3 × 7 and 4 × 7 Youden squares; these 29 isotopy classes belong to 19 of the 147 main classes (species) of 7 × 7 Latin squares. We also give some related results on partitioning 7 × 7 Graeco-Latin squares.