An autotopism of a Latin square is a triple (a,b,c) of permutations such that the Latin square is mapped to itself by permuting its rows by a, columns by b, and symbols by c. Let Atp(n) be the set of all autotopisms of Latin squares of order n. Whether a triple (a,b,c) of permutations belongs to Atp(n) depends only on the cycle structures of a, b, and c. We establish a number of necessary condi...

Late in his long and productive career, Leonhard Euler published a hundred-page paper detailing the properties of a new mathematical structure: Graeco-Latin squares. In this paper, Euler claimed that a Graeco-Latin square of size n could never exist for any n of the form 4k+2, although he was not able to prove it. In the end, his difficulty was validated. Over a period of 200 years, more than t...

A Latin square is pan-Hamiltonian if every pair of rows forms a single cycle. Such squares are related to perfect 1-factorisations of the complete bipartite graph. A square is atomic if every conjugate is pan-Hamiltonian. These squares are indivisible in a strong sense – they have no proper subrectangles. We give some existence results and a catalogue for small orders. In the process we identif...