Three and a Half Trillion Sudoku Puzzles

C|Net news.com has a nice column that points to this American Scientist article about the game of Sudoku. The article begins with a nice diatribe about the fact that mathematics is more than arithmetic, and continues on to expose (page 4) the following:

...9 x 9 Latin squares were not enumerated until 1975; the tally is 5,524,751,496,156,892,842,531,225,600... The order-3 Sudoku must be a subset of these squares. They were counted in June 2005... 6,670,903,752,021,072,936,960...

{Skip Ahead a Bit, Brother}

When all these symmetries are taken into account, the number of essentially different Sudoku patterns is reduced substantially. In the case of the order-2 Sudoku, it turns out there are actually only two distinct grids! All the rest of the 288 patterns can all be generated from these two by applying various symmetry operations. In the order-3 case, the reduction is also dramatic, although it still leaves an impressive number of genuinely different solutions: 3,546,146,300,288...