Construction of bimagic squares using orthogonal Sudoku squares (extended abstract)

It is well-known that Tarry was the first to prove that orthogonal latin squares of order six do not exist. Less well-known is that he was the first to give constructions for bimagic squares valid (in theory) for all orders p2, where p is a prime. He used in effect pairs of orthogonal diagonal Sudoku squares, the ones used for a particular prime p being determined by an appropriate “key”. [A square is bimagic if it is a magic square and remains magic when all its entries are replaced by their squares.]We show that one of the pairs of orthogonal diagonal Sudoku squares which is appropriate for the construction when p=3 can be generalized to provide a standard construction valid for all primes p except p=5. We explain why the construction fails when p=5.