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Comment by ddlatham | Hacker News Reader

Is it fair to rely on the fact that there is only a single solution to make deductions?

Yes. Any deduction that you could reach from that fact could also be reached from other constraints on the board.

Here's an example illustration, from the top-left corner of a board:

  y B R
  x R _
  _ _ _

If x is red, then y is indeterminate, either red or blue is valid there, no other constraint can "see" y to determine it. But if we know the puzzle has a single solution, then we can correctly say x must be blue, so that some other constraint south of x can see through to y. This could be worked out later from that other constraint. But the rules yield this emergent property of a shortcut, and any such emergent property is fair game for logical deduction.

If there's anything unfair about this approach, it's that the existence of a single solution is merely implied by puzzle conventions and not explicitly stated as a rule for 0h n0 here.

There's actually a constraint that no blue circle can be alone, which you can find by reaching an end board with a solitary blue that the game will prompt you about. So X red and Y blue would be well-determined.

However there does appear to be another constraint that is not mentioned explicitly, which is that there are no blue circles that are not in line-of-sight with a numbered circle.

That last isn't defined explicitly. It's an emergent property from the constraint that the solution must be unique.

Any blue circle (or any linear group of them) that is not in line-of-sight with a numbered circle would have no constraint preventing it/them from being red instead. Therefore the solution wouldn't be unique, so such blue circles can't exist.

Some starting conditions have multiple solutions. A trivial example would be a 2 x 2 game with two "2" at opposite corners.

I believe that such instances are not generated by this game, however.

I had an example with two possible solutions (3 empty spaces were left, and either BRB or RBR would have satisfied the conditions). I meant to take a screenshot after finishing but the final position fades out (and adds numbers everywhere) so I should have done it before the end.

I would give an unqualified yes to this. All rules and constraints of a puzzle are fair game for making deductions.

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