Symmetry Groups of Multiple Gerechte Designs

Abstract

A gerechte design is an nxn Latin square together with a “skeleton” which partitions the square into n additional regions, each containing the symbols 1...n. For example, Sudoku is a gerechte design in which the additional skeleton consists of nine 3x3 squares. The symmetry groups and corresponding equivalence classes of certain gerechte designs, such as Sudoku and its smaller version Shidoku, have been well studied as a method of counting unique solutions and for their use in designing agricultural experiments. We extend these results to designs with non-square (contiguous and identical) regions. In particular, we study 4x4 multiple gerechte designs, i.e. gerechte designs satisfying multiple skeletons. Given a set of gerechte designs sharing one or more skeletons, we examine the relationship between the symmetry group of this set and what we call pairwise variation, a measure of the randomness of the given designs.