Enumerating possible Sudoku grids

In some of these boxes, the setter puts some of the digits 1–9; the aim of the solver is to complete the grid by filling in a digit in every box in such a way that each row, each column, and each 3× 3 box contains each of the digits 1–9 exactly once. In this note, we discuss the problem of enumerating all possible Sudoku grids. This is a very natural problem, but, perhaps surprisingly, it seems unlikely that the problem should have a simple combinatorial answer. Indeed, Sudoku grids are simply special cases of Latin squares, and the enumeration of Latin squares is itself a difficult problem, with no general combinatorial formulae known. Latin squares of sizes up to 11 × 11 have been enumerated, and the methods are broadly brute force calculations, much like the approach we sketch for Sudoku grids below. See [1], [2] and [3] for more details. It is known that the number of 9× 9 Latin squares is 5524751496156892842531225600 ≈ 5.525× 10. Since this answer is enormous, we need to refine our search considerably in order to be able to get an answer in a sensible amount of computing time.