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Why must a sudoku have a unique solution. In my books I don't guarantee it. But I am sure there are not many which don't. I doubt if one in a thousand dont have. And even if they don't its only a matter of interchanging about four numbers. Why is there this obsession that it has to be only one unique solution. This of course applies to other puzzles too.

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closed as unclear what you're asking by Kendall Frey, Doorknob, Aza, Ross Millikan, Joe Z. Jun 2 '14 at 21:51

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  • $\begingroup$ Why don't they make jigsaw puzzles so that every piece connects with every other piece? What's the point of a puzzle that you have to make up a solution? $\endgroup$ – Kendall Frey Jun 2 '14 at 13:00
  • $\begingroup$ You still have to decide a correct solution. A puzzle (in almost all cases) is something that you have to find the solution, not create your own. $\endgroup$ – Kendall Frey Jun 2 '14 at 13:08
  • $\begingroup$ Actually it doesn't. Although most sudoku you find will usually have a unique solution it is not fully true. It is possible to find ones that does have more than one solution. All this is vague as Sudoku doesn't come with official rules as to what exactly constitutes a Sudoku. $\endgroup$ – minusSeven Jun 2 '14 at 13:08
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    $\begingroup$ In the case of Sudoku, it was originally intended to be a puzzle to be completely filled out using nothing but logic. If multiple solutions exist, this defeats the purpose, as you can't complete it with logic alone. $\endgroup$ – Kendall Frey Jun 2 '14 at 13:15
  • $\begingroup$ @AlanRoss Is this an actual question, or do you just want to argue that puzzles with multiple solutions are valid puzzles too? $\endgroup$ – SQB Jun 5 '14 at 10:23
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There is no law requiring that a published Sudoku have a unique solution. When I see a puzzle of any type, I expect from experience that the setter has promised a unique solution (or occasionally will say there are some number to be found). Some setters, Raymond Smullyan especially, create problems that challenge you to make use of the fact that there must be a unique solution. Cryptarithms are especially prone to multiple solutions.

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    $\begingroup$ I understand what you are saying. I think most Sudoku solvers would be disappointed to find there were multiple solutions. It is quite possible to remove four numbers and have two solutions. I believe most Sudoku setters run the final product through a solver to make sure there is a unique solution. They certainly do not have a fixed number of clues. You can do what you want-I was saying what I believe is the expectation of most of the puzzling community. $\endgroup$ – Ross Millikan Jun 2 '14 at 14:01
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Many of the advanced solving techniques rely on there being a unique solution. All of the Unique Rectangle techniques are predicated on this. To quote from that page:

Uniqueness Techniques are based on the fact that practically every Sudoku sudoku ever published has only one solution. More than that: Most Sudoku players consider sudokus with more than one solution invalid!

I would say that uniqueness is the default convention, in that unless you explicitly say otherwise, everyone will assume that any Sudoku puzzle has a unique solution, and failure to say that a given puzzle may not have a unique solution could result in unhappy customers.

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  • $\begingroup$ From that article: "Most Sudoku players consider sudokus with more than one solution invalid!". I would want evidence to quantify "most". +1 though for stating the link between "default convention" and "happy customers". $\endgroup$ – ClickRick Jun 22 '14 at 8:32
  • $\begingroup$ Well, as evidence I present all of the sites that mention uniqueness techniques (such as unique rectangles). ALL of these techniques rely on puzzles having unique solutions. $\endgroup$ – Donald.McLean Jun 22 '14 at 16:29
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You can solve Sudoku with different approaches. For instance, you can fill in all blank squares with random digits and see if the result meets the Sudoku 1-9 digit-set requirements; if not, try some other random digits. This is a trial-and-error approach. Most people, though, want to use deduction based on the digits previously established in puzzle squares.

If a Sudoku has multiple solutions, deduction will fail to solve the puzzle at some (late) point. The partial solution sequence will reach a point where no empty square can be deduced as having to be a specific digit. For people who do not use guessing (trial-and-error), this can be a frustrating barrier (probably making them curse the puzzle maker).

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  • $\begingroup$ A Sudoku with 81 empty squares can also be solved logically; you just fill in digits that don't conflict. If a digit to be added next logically conflicts with all possible remaining permutations, then it is not logical to add it; use a different digit. This is the same sort of logic, just carried back to 81 empty spaces instead of only 4 empty spaces. $\endgroup$ – mgkrebbs Jun 3 '14 at 0:00
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    $\begingroup$ I'm sorry, @AlanRoss, but your argument is completely meaningless. A "solution" is, by most people's accepted definition, unique. You don't have to accept their definition, but trying to tell them they are wrong just because you disagree accomplishes nothing positive. $\endgroup$ – Donald.McLean Jun 3 '14 at 0:11

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