AI-assisted proof of optimal packing for 11 squares

8 comments

• yzydserd 1 hour ago
  fwiw The prime site for square in square packing is at https://kingbird.myphotos.cc/packing/squares_in_squares.html

  The triangular view is most interesting. And a 20 minute video on this view is at https://youtu.be/uL5wuiy34rs

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  • Buttons840 45 minutes ago
    "God is dead and the optimal packing of squares killed him." I will never not think of this meme when looking at these horrors. I see it, but I don't like it. ;)
  • woah 22 minutes ago
    Can someone explain why 83 and 87 can't get any smaller?
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    • entropicdrifter 2 minutes ago
      Because the outer perimeter must be a square. 83 and 87 could shrink the outer perimeter in one dimension, but not in both at the same time.
    • danbruc 6 minutes ago
      Which of the blocks do you think you could move to shrink the solution? Or are you thinking of a completely different arrangement?
    • nemomarx 7 minutes ago
      They got updated to be smaller this year, so maybe there's still more gains to be had?
  • schiffern 46 minutes ago
    More on the 11-squares packing:

    https://startupfortune.com/ai-models-formally-proved-walter-...

    https://vplevris.medium.com/eleven-squares-one-tiny-gap-and-... (written just days before the new proof!)

    https://jlevy.github.io/squares/cases/11.html
• WithinReason 1 hour ago
  A list of many square packings, with images:

  https://jlevy.github.io/squares/

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  • aunty_helen 1 hour ago
    I like geometry. These packings show there are ugly numbers, like 51.
• agnishom 2 hours ago
  The readme has no figures :( describing the packing?
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  • fredsted 1 hour ago
    There's a cool figure here: https://x.com/ojoshe/status/2107590622005924265
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    • wackget 1 hour ago
      Any mirrors which don't require giving clicks to neo-Twitter, please?
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      • schiffern 1 hour ago
        https://kingbird.myphotos.cc/packing/squares_in_squares.html

        For more packings (circles in circles, etc) check out this page: https://erich-friedman.github.io/packing/index.html
  • tantalor 1 hour ago
    Pics or it didn't happen
  • vessenes 2 hours ago
    I had the same thought! Pics please.

    EDIT: I found it a few links down. https://jlevy.github.io/squares/cases/11.html

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    • jo-han 1 hour ago
      <https://en.wikipedia.org/wiki/File:Packing_11_unit_squares_i...> from https://en.wikipedia.org/wiki/Square_packing
• mlmonkey 1 hour ago
  Lot more pics here: https://jlevy.github.io/squares/
• coppercrisp62 1 hour ago
  Did an interval-arithmetic branch and bound once, getting the rounding modes right took me weeks.
• reader9274 47 minutes ago
  Another interesting video related to these types of problems: https://youtu.be/mVH7OPx4QZU
• rfgplk 1 hour ago
  Isn't this obvious? Why do you need a proof for it, just stack the cubes next to each other? If we're talking infinitesimally thin squares, then stack them on top of each other? Am I missing something?
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  • raincole 49 minutes ago
    It takes less time for you to try to read about the question than to type this comment. I know the link doesn't contain visualization, but... come on.
  • 233mhz 55 minutes ago
    The whole point is that you can fit more than by naively stacking them...
  • AlexandrB 53 minutes ago
    Look at some of the other links people have posted for optimal packings. The optimal 11 square packing looks nothing like what you're describing ("just stack the cubes next to each other").