## Saturday, August 3, 2013

### Self-Referential Puzzle, by Jack Webster

If you've been following our Facebook page, you know that Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers is nearing the end of copy editing. We still need to add some artwork and get the pages laid out nicely. Then we'll be done.

Today I'm working on solutions to all of the puzzles in the book. I just spent two or three hours lost in the craziest puzzle. I had solved it a few years ago when I first saw it, but of course I don't have notes from back then.

If I hadn't solved it before, I'm not sure I would have believed I could do it. When you first look at it, it doesn't seem possible that the few clues Jack gives could be enough. They are. I have a page-long explanation of the logical steps I took to solve it. I wonder if it's possible to solve it in any other order.

If you like challenges, this is a good one. And if you like this, Jack has others on his site.

1. I love, love, LOVE puzzles! Thanks for the new source, Sue.

2. Shireen, did you see the reindeer puzzle I posted in December? It's also very challenging, and I found it delightful.

3. I didn't link it before, but now it's on my Reindeer Radar! Thanks for more fun. I had fun with this puzzle this afternoon and printed out some more for later :).

4. re: number of clues; This reminds me of some puzzles that appear on the USPC. They make instructions available the day before with sample puzzles, but the puzzle designers often like to create the actual puzzle slightly different from the sample. (For instance, the sample might be on a rectangular grid [the form of the puzzle is not specified in the written instructions], and the actual puzzle will be on a tesselation with pentagons, hexagons, heptagons, etc.)

5. Lovely puzzle! Interestingly, as sparse as the clues appear to be, they still contain some redundant information. After solving it by hand, I wrote a brute-force program to check whether there was more than one solution. But I was lazy and left out a few of the constraints that were going to be harder to check. That program produced three solutions, and putting back one of the missing constraints was enough to make the solution unique. I haven't gone back to solve the problem from scratch *without* the missing constraints, so I can't say how they would affect the experience of solving the puzzle.

6. Fascinating! Which two constraints could be left out? I sure used every clue there in order to solve it.

7. ***POTENTIAL SPOILER***
From the E clue, I used that there were only two 2's but I did not program that cell E was equal to the horizontal and vertical distance between the 2's. Originally, from the O clue, I used that there was a single 7, but did not program the constraint on the column of the 7. I had to put the constraint on the column back in to get a unique solution.

8. I don't think that will be much of a spoiler. ;^) I definitely used the distances between the 2's in my solution. It's intriguing to know that it would be possible to solve it without knowing that.

9. Hi Chris,

That's really interesting. So to be clear, if we change clue E to 'There are only 2 Es' there is still a unique solution? That seems quite extraordinary to me as the rest of the clue is quite important for me when solving it (at least the way I solved it; I'm sure there are others).

I'm really glad you enjoyed it though. This might be a chance to improve it a bit though - I've been slowly improving it over the years. (The other three I haven't changed so much)

10. Jack, yes, there is still a unique solution (modulo the typo). And I agree -- when I solved it by hand, that clue was also very important for me!

11. This is awesome!