Is solving a puzzle using uniqueness invalid? [duplicate]

Question

(I don't know if this is appropriate to ask here, since it's a bit of a soft question about puzzles, rather than a puzzle itself. I'm not certain about the tags either.)

When I was first introduced to Sudoku, I was told, in no uncertain terms, that there had to be one and only one solution. The way I took that to mean is that any 9 by 9 grid with numbers 1 to 9 in it, that could be completed in multiple ways (or no ways at all) was not a Sudoku puzzle (maybe just a cruel joke to play on puzzlers).

Since then, I've found a whole bunch of similar kinds of puzzles, and often the same rule carries over: there can only be one solution. Even if it's not explicitly stated, people still feel cheated if there is more than one.

But, whenever I see people doing these kinds of puzzles, the techniques tend to ignore this requirement. They're usually quite direct, of the form "in order for there to be even one solution, we must have ....". I just don't see many techniques of the form "this can't be this way, or else this will cause at least two solutions".

It's not that these techniques can't be helpful. I have had experiences in the past where I haven't been able to crack a problem by these "existence" methods, but I can identify situations that would break uniqueness.

Here's a concrete example of a uniqueness method. Take the puzzle found here (you can play online). You have an n by n grid, filled with the numbers 1 to n. You are supposed to black out numbers in the grid, until the grid satisfies the following:

1. Each number shows up, without being blacked out, at most once.
2. No two adjacent (orthogonally, not diagonally) squares maybe blacked out.
3. The remaining white space must be connected (again, when moving orthogonally, not diagonoally). That is, you cannot "wall off" certain white spaces with diagonally connected black squares.

and of course, there is only one solution.

Here's a rule we can deduce from uniqueness that cannot otherwise be deduced: if there is a number that does not have any duplicates in the row and column in which it is found, then that number cannot be blacked out.

Why? Because we can always elect not to black out a number, and the only rule violated is rule 1. If no duplicate number appears in the row/column, then even rule 1 cannot be violated. So, we have another, distinct solution, which is not acceptable. Therefore, the number could never be blacked out in the first place.

From there, you could even go further! You could say that either an orthogonally adjacent square must be blacked out, or at least two diagonally adjacent square must be blacked out.

You'll find this stuff out eventually with other methods, but that's not the point. When things get really difficult, we sometimes need to take progress when we can get it, and if uniqueness methods help, then why don't we use them?

tldr: Many puzzles require that their solution be unique. Why don't more people use this fact to solve them?

Answer 1

Deducing backwards from the assumption of uniqueness does not prove uniqueness. In other words, if you use uniqueness as the premise, the best you can ever find is "a solution".

On the other hand, if you deduce from first principles only, any solution you find will be "the solution", because you'll have ruled out any other solutions on the way.

It's entirely up to taste, circumstances (and, of course, the puzzle creator), which kind of solutions are better. Some puzzle types (e.g. numberlink) have interesting insights and complex techniques that you can only use if you can assume uniqueness, so it's a very good idea to make those puzzles explicitly unique, and accept the answer without any further uniqueness considerations.

On the other hand, some maths puzzles (e.g. this one) will lose all their appeal if you deduce the answer from uniqueness, because the point was to show that there's a clever way to prove that uniqueness. In these cases, I'd go as far as to say that deducing from assumed uniqueness is a mistake.

Sudoku puzzles stand somewhere in the middle ground, in my opinion. Super advanced sudoku addicts may like puzzles where you have to choose between guess-and-brute-force methods, and deducing backwards from assumed uniqueness. I find plenty enough challenge for myself even without such shenanigans, so I tend to like my morning sudoku better, if neither of those methods are needed.

(I haven't tried the puzzle OP used as an example, so I may be wrong in thinking that OP's uniqueness method looks a lot like it was intended by the puzzle creator.)

TL;DR: Deducing from assumed uniqueness is sometimes ok, sometimes a mistake, and always subject to the consideration of "which way does it make for a better puzzle".