On the one hand, there is only one fundamental rule to Sudoku:
You must place digits into the grid in such a way that every row, every column, and every 3x3 box contains each of the digits 1 through 9.
There is nothing in this rule that states the puzzle must have a unique solution. Indeed, the extreme case--a blank puzzle, with no givens whatsoever, has over 46,000 solutions just from possible placements of a single digit on the starting blank puzzle.
So there is nothing in the rules, some say, that entitles a solver to assume a Sudoku has a unique solution.
On the other hand, say others, nearly all published Sudoku puzzles have only one solution--or claim the solution is unique. It is irritating both to the publishers of such puzzles and to their solving audience if the puzzle has more than one solution. So if you know that the proposer of the Sudoku intends it to have a unique solution, why shouldn't you be allowed to take advantage of that bit of knowledge when making logical deductions to solve the puzzle?
Again on the other hand, the claim has been made that all logical deductions made by the Uniqueness family of solving techniques that assume a puzzle has a unique solution can be found by other means that do not require the Uniqueness assumption.
But again contrariwise, no rigorous mathematical proof of this assertion has been put forward.
And so the controversy continues.
The following should help clarify and limit the controversy:
- Sudoku axioms are constraints on the solution, given the entries (whichever these are); they are constraints the PLAYER must satisfy;
- Uniqueness is a constraint on the entries; it is a constraint the PUZZLE CREATOR must satisfy and the player may use if he trusts the puzzle creator; for the player, it is like an oracle on the puzzle.
Accepting rules based on the axiom of uniqueness is thus mainly a matter of personal taste. There is nevertheless an objective feature of these rules: even if they are not needed (which is not proven), they may simplify the solution.
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