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I think this question is best explained with a demonstration. In this case I'm playing a Range puzzle, but this kind of concept seems to work with other puzzle types too.

image of partially solved Range puzzle

Of interest here are the two blank squares in the last row of the puzzle. Let's call the left of the two blank squares A, and the right B.

If square A is black, then square B must be white, since two black squares cannot be orthogonally adjacent.

If square A is white, then square B may be white or black, and the puzzle would still be in a valid state. No other restrictions are placed on square B.

Therefore, square A must be black, to guarantee a unique solution to the puzzle.

My question is, when solving puzzles, is this kind of logic sound? If I follow this logic through in other contexts, is it always guaranteed to produce a correct conclusion? Is there a name for this kind of logic?

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This is generally called uniqueness logic -- assuming that the puzzle has a unique solution, and using that to find the solution. This is sound if the puzzle does indeed have a unique solution, but if it doesn't (because of some error by the constructor/generator), you're out of luck.

In human-designed logic puzzles, this type of thing is generally considered bad. The guarantee "each puzzle has a unique solution" is taken to be outside of the rules of the genre. Puzzles are designed so that you shouldn't need to use uniqueness logic: there is a 'nice' solving path without using it. (After all, in order to make sure that the puzzle did have a unique solution, the constructor couldn't use it!)

So this type of logic may be sound, if the constructor's done their job right. But it's outside of the "spirit" of the puzzle.

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It really depends on the type and level of the puzzle.

If you're solving a puzzle of a type where you're explicitly told that there should be a unique solution, then you can use that information in the way you're doing. Not all puzzles are guaranteed to have unique solutions: I guess all good puzzles should, but sometimes a home-made puzzle posted here on Puzzling SE may not be so perfect, and often it's enough just to find a solution without proving that it's unique.

Usually you won't need to use such "meta" deductions, and it's enough to use more direct logic to find out what should be in each square. After all, if the puzzle does have a unique solution, then the non-unique possibilities you've found must lead to a contradiction somewhere, not just by being non-unique. (In your example, there can't be a valid solution with square A being white - so making square A white must necessarily lead to a contradiction somewhere else, otherwise it would be a valid solution!)

If the puzzle has a unique solution, then this type of deductive step isn't logically necessary. (And if it doesn't have a unique solution, then of course this type of deductive step is invalid!)

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