In this particular puzzle it’s not actually necessary to apply the uniqueness constraint. There is at least one other pathway to a solution—which I’ll leave to you to find.

Your sequence of moves is the pathway I had in mind.

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]]>In this particular puzzle it’s not actually necessary to apply the uniqueness constraint. There is at least one other pathway to a solution—which I’ll leave to you to find.

Your sequence of moves is the pathway I had in mind.

]]>If we use Excel labels, top row is 1, second row 2, etc., and left column A, next column B, etc., then cells E1 F1 G1 must contain numbers 1 and 2 and 3 somewhere. The region F2:F6 has 5 cells, 1 and 5 have been decided. F2 could be 2 or 3 or 4, except that we just eliminated 2 and 3 from consideration. F2 must be 4. We continue and decide F3 must be 3 and F6 must be 2.

Next consider the region E2:E7. Given certain cells have been decided, E5 could only be 4. This means D4 cannot be 4, which means D3 must be 4, leaving D4 to be 2.

My final solution is this, without relying on the unique solution rule.

1 3 2 5 1 3 2

4 5 1 3 2 4 5

1 2 6 4 1 3 1

3 4 3 2 6 5 2

2 1 5 1 4 1 3

6 3 4 6 3 2 4

5 1 2 1 5 1 6

But if there are no solutions, the puzzle solver should be required to find a proof of that fact… of course, the nice thing about puzzles is that the “certificate” of a solution is the solution itself, which fits on the page, while a “certificate” of unsolvability might be exponentially large. In CS terms, coNP and NP are probably different.

]]>But I do still wonder about the practical safety of such schemes. When the interrogator demands the secret key and threatens you with the rubber hose, you have to consider the possibility that he too knows about deniable encryption, and may not be satisfied with an innocuous message, whether or not it’s the correct decryption. In the case of the one-time pad, where *any* decryption is possible, the interrogator can just make up his own incriminating key.

As for the puzzle that punishes you for assuming it has a unique solution, I think you can create such a beast from the Capsules puzzle shown above by changing two of the givens. As you say, it’s a cute trick. But if I encountered it in the *Times*, I would write a complaining letter to the puzzles editor.

Place numbers in the grid so that each outlined region contains the numbers \(1\) to \(n\), where \(n\) is the number of squares in the region. The same number can never touch itself, not even diagonally.

the setter would be within their rights to make a puzzle *without* a unique solution. So one can’t use the uniqueness of the solution, because that’s not a given.

But of course it is always a valid approach to guess the contents of some square and the proceed until one solves the puzzle or reaches a contradiction. So you can still solve the puzzle using exactly the same steps as you would have done using the uniqueness assumption. It’s just that you don’t think

Since the puzzle has a unique solution, those squares must be \(2\) and \(4\).

but rather

Since I suspect the puzzle has a unique solution, I will proceed on the assumption that those squares are \(2\) and \(4\) and see where that gets me.

while you proceed to put exactly the same numbers in exactly the same squares.

I wonder if one could make a fun type of puzzle which didn’t have a unique solution. One could arrange matters so that *most* guesses eventually lead to a contradiction, so that the skill would be in penciling in numbers which *were* part of a global solution. It would have a rather different feel to it than most puzzles.

I don’t know if anyone has used this in practice, but it seems interesting…

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