How many combinations are possible in a 9X9 Sudoku matrix?
arvernus: Enumerating the Sudoku 9×9 grid solutions directlyThe first approach taken historically to enumerate Sudoku solutions (Enumerating possible Sudoku grids by Felgenhauer and Jarvis) was to analyze the permutations of the top band used in valid solutions. Once the Band1 symmetries and equivalence classes for the partial grid solutions were identified, the completions of the lower two bands were constructed and counted for each equivalence class. Summing completions over the equivalence classes, weighted by class size, gives the total number of solutions as 6,670,903,752,021,072,936,960 (6.67×1021). The value was subsequently confirmed numerous times independently. The Algorithm details section (below) describes the method.(http://en.wikipedia.org/wiki/Mathematics_of_Sudoku#Enumerati...)