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3 | added 877 characters in body | ||
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Or if you want something that looks like the puzzles that get published, here's another one that uses the same mechanism: ----------------------------------- | 1 . . | 8 . 7 | 5 . . || | | || . 5 . | 6 . . | 9 7 . || | | || . 6 . | . 4 . | . . . || . . 2 | . . . | . 6 . || | | || 6 . . | 4 . 3 | . . . || | | || . 4 . | . . . | 1 . . || . . . | . 1 . | . 5 . || | | || . 1 3 | . . 6 | . 9 . || | | || . . 7 | 5 . 4 | . . 1 | ----------------------------------- |
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2 | add example puzzle | ||
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ETA: It does indeed seem to be true that these rules don't imply unique rational solutions. With them, my solver easily solves the following puzzle, which has a unique integer solution but does not have a unique fractional solution. Without them, my solver can still solve the same puzzle, but only by using rules that are (in my experience) much more difficult to apply by hand. ----------------------------------- | 3 7 8 | 6 4 5 | 1 2 9 || | | || 6 9 . | . 7 . | 4 8 5 || | | || 4 . 5 | 9 . 8 | 3 7 6 || 7 . 9 | 5 3 . | 8 6 4 || | | || 8 4 . | 7 . . | 5 9 3 || | | || 5 3 6 | 8 9 4 | 2 1 7 || 1 6 3 | 2 5 7 | 9 4 8 || | | || 9 8 4 | . . . | 7 5 2 || | | || 2 5 7 | 4 8 9 | 6 3 1 | ----------------------------------- |
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I wasn't planning on answering here but since someone mentioned my paper in the long comment thread above maybe I should anyway. When I'm solving problems by hand one of the sets of patterns I frequently use involve uniqueness: something can't happen because it would lead to a puzzle with more than one solution, but a well-posed Sudoku puzzle has only one solution, so it's safe to assume that whatever it is doesn't happen. For instance, it's not possible to have four initially-empty cells in a rectangle of the same two rows, the same two columns, and the same two 3x3 squares, and also for these four cells to all contain one of the same two values, because then the two values they contain could be placed in two different ways and the rest of the puzzle wouldn't notice the change. So if there's only one way to prevent a rectangle like this from forming then the solution has to use that one way. I have the sense (though I could be wrong) that this sort of inference does not imply a unique rational solution. But of course a puzzle could have a unique rational solution anyway — my computer solver knows these rules too, but uses them less frequently than I do by hand. |
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