Gary McGuire writes: << Call two completed sudoku grids "equivalent" if one can be obtained from the other by a sequence of the following operations: 1) permutation of the digits 1-9 2) permutation of the rows. These come in two types: a) permute the three rows within a given band b) permute the bands 3) permutation of the columns, same two types. 4) rotation, reflection
This makes me realize that my previous attempt to define an equivalence relation on Latin squares was much too weak. (I'm more interested in Latin squares than Sudoku grids with their kind-of random condition on certain 3x3 squares.) Call two of these equivalent if they become identical by some combination of the following: 1) any permutation of the n symbols 2) any permutation of the n rows 3) any permutation of the n colums 4) application of any dihedral element from D_4. This suggests most equivalence classes have 8(n!)^3 elements (but not necessarily that many since some combinations of these equivalencing operations may not change anything). Problem: How many n by n Latin squares are there before taking equivalence classes? How many equivalence classes are there? --Dan