From The Importance of Automorphic Puzzles

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`6,671,248,172,291,458,990,080 5,472,730,538 * 3,359,232 * 362,880`

6,670,903,752,021,072,936,960 exact number of Sudoku solution grids

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344,420,270,386,053,120 grid deficiency

5,472,730,538 exact number of unique Sudoku solution grids

3,359,232 possible geometric permutations of solution grid

362,880 number of ways to renumber a grid

Given row 1 = 123456789, then there are, I think, 150 possible settings for column 1 that produce canonical (minlex) solutions. The first setting is:

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`1 2 3 | 4 5 6 | 7 8 9`

4 . . | . . . | . . .

5 . . | . . . | . . .

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2 . . | . . . | . . .

3 . . | . . . | . . .

6 . . | . . . | . . .

---------------------

7 . . | . . . | . . .

8 . . | . . . | . . .

9 . . | . . . | . . .

And the last (in lex order) is:

- Code: Select all
`1 2 3 | 4 5 6 | 7 8 9`

8 . . | . . . | . . .

9 . . | . . . | . . .

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2 . . | . . . | . . .

6 . . | . . . | . . .

7 . . | . . . | . . .

---------------------

3 . . | . . . | . . .

4 . . | . . . | . . .

5 . . | . . . | . . .

My question (and I sense the possibility of public humiliation here but what the heck!) is this - if I count all the solutions for each of these 150 grid patterns, shouldn't I get 5,472,730,538?

The first handful of col 1settings that I tested produced these numbers:

- col 1 = 145237689, solutions = 744,603,884

- col 1 = 145238679, solutions = 805,608,877

- col 1 = 145239678, solutions = 872,751,263

- col 1 = 145267389, solutions = 1,336,808,742

- col 1 = 145268379, solutions = 1,358956,881

That's nearly 5 billion in just the first 5 grid patterns, so clearly there is something fundamentally wrong with my approach. WTF??