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Dear Reader: This question assumes you know what a Nonogram is. If you do not, I recommend reading the Wikipedia entry first.

Nonograms have many solving techniques. One of the lesser used ones - simply because it does not come up that often - is that if the puzzle is symmetric, you can do additional solving techniques; specifically, for a particular row - if the nonogram is vertically symmetric - the center number must be centered in the middle. Also, any solving you do on one side of the puzzle can also be immediately copied to it's reflection on the other side.

This question is about diagonally symmetric Nonograms. You can tell if a puzzle is diagonally symmetric (top left to bottom right) if the numbers on each side are the same, and symmetric top right to bottom left in the numbers are the same, but in opposite order. Here is an example of a diagonally symmetric Nonogram:

Basic Nonogram Nonogram with Symmetry line

As you can see, the numbers are the same, but in reverse order, so it has top right to bottom left symmetry. Are there additional solving techniques that take advantage of this fact? How would you solve this (fairly difficult for a 10x10) puzzle?

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  • $\begingroup$ I think that example is only symmetric if you flip the "1, 4" clue in the fifth row to "4, 1". $\endgroup$ – Kevin Feb 11 '15 at 15:21
  • $\begingroup$ @Kevin I finally had time to fix the images. Had a busy day at the office :) $\endgroup$ – durron597 Feb 11 '15 at 21:08
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Any symmetrical puzzle will be mirrored over the axis of symmetry. It behaves that way for any symmetrical Nonogram, I would expect, whether it's vertical symmetry, horizontal, or diagonal. I'm not sure about there being any extra tactics toward solving it that couldn't be applied to an asymmetrical puzzle, other than the ones you mentioned in the original post. The way I see it, if the tactic of mirroring your actions over the axis of symmetry doesn't work, then it's either not a symmetrical Nonogram or it's an invalid puzzle.

Here's the solution to your 10x10, and as you can see, it is indeed symmetrical over the diagonal line in your description. If you'd like me to explain in detail how I solved it, I can, but only when I have time after classes today:

Nonogram

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  • $\begingroup$ This is not really the sort of answer I was looking for. The puzzle I gave was just an example to give the post a little flavor, I'm asking more about techniques that capitalize on the symmetry aspect. FWIW I solved this one by guessing that the bottom left corner was filled and that let me deterministically solve the rest of the puzzle. I wasn't really happy with that answer, though. $\endgroup$ – durron597 Feb 12 '15 at 21:46
  • $\begingroup$ Right, I figured this wasn't the answer you were looking for. Unfortunately, there might not be an answer, either. The answer I gave was that the same rules will apply, and I cannot expect there to be any other special rules that would exist for a diagonal symmetry than any other kind of symmetry, or lack thereof. $\endgroup$ – Bulldogg6404 Feb 12 '15 at 21:49
  • $\begingroup$ I guess it would be very hard to prove that "no additional strategies exist" even if that were the case. $\endgroup$ – durron597 Feb 12 '15 at 21:50
  • $\begingroup$ Additionally, I guess that any other strategies would not be simpler than the ones already given. Proper Nonograms by themselves are never all that difficult, and having symmetry in any form makes it all the more easy with the strategies you know already. The more important question is probably identifying lines of diagonal symmetry before you start, which I give credit to Kevin on that, being able to spot the 1/4 mix-up without (assumption) attempting to solve it. $\endgroup$ – Bulldogg6404 Feb 12 '15 at 21:59

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