Why this sliding tile puzzle arrangement is impossible

boing
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Originally published at: https://boingboing.net/2020/04/21/why-this-sliding-tile-puzzle-a.html

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I love numberphile. It’s up there with 3Blue1Brown and Mathologer. @frauenfelder if you’re not familiar with those other channels, you might want to check them out, too.

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Video link for the BBS
https://www.youtube.com/watch?v=YI1WqYKHi78

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Oh my yes. Love the FT stuff on there, and use it for my upper level class every year.

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Think outside the, er, box, just pop them suckers out! :wink:

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I see you’ve discovered my solution to the Rubik’s Cube.

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My 11 year old is vindicated. He, 9yo and I have been trying for the last three days to put one of these puzzles in reverse numerical order and cannot get the 2 and 1 in the right order. 11yo deemed it impossible. I will play this for them tomorrow as part of math distance learning.

Edit for grammar.

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Way back when I used to make (unobtrusive, selfhosted) Flash ad’s I had a run in with this type of puzzle. We decided the submit form for “leave your email address to win tickets to the first transformers movie” would only be available after you solved a slide puzzle of one of the robots. (We didn’t even force people to sign up for the newsletter in order to participate, different times.)

I had only rudimentary Flash skills, basically no programming skills. So I went looking for existing code. Found a code block with a permissive license and took it for a test run. A few hours later, and many many attempts later, I figured out that:

  1. This code shuffled the pieces at random.
  2. Because of that roughly 50% of the time the puzzle was not solvable.

I intuitively understood much of what the video was saying, ie: if the last two pieces have to be swapped, while the rest is correct, that is impossible. Really nice to know the full story behind that.

The other lesson I learned that day. If you tell your boss he has to pay a few bucks in order to save you a few hours of time, that is often a good deal for them. I ended up buying, with the company credit card, a element that was much nicer and that didn’t randomly shuffle.

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Comparing the photo with the still from the video, it’s obvious why it’s impossible: Reds are odd in the photo, and they are even in the video. No amount of sliding will change that. DUH.

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I always did that. Not exactly difficult with the cheap plastic ones.

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