I'm working on a reproduction of the proof with some personal changes. The basic approach is the standard computer assisted "unavoidable set" approach. First, choose some regions small enough that two squares don't fit, the article used 16. Each region must contain or not contain a square, which is 16 choose 11 cases, about 2000. For each case you try and rule it out. You do this by identifying regions that must be covered by a square, and propagating this information. You can also use packing LPs like Stromquist did in 1989 to rule out more configurations. You then narrow in on the remaining cases and subdivide them more.<p>I think the only reason this wasn't done pre-AI was due to it not being a topic of serious focus. 1989's computers were too weak to handle all the cases. But all the basic ingredients were present in the Kepler conjecture proof. What AI did was lower the effort enough that amateurs who just liked square packings could perform and formally verify such a proof. I consider myself among such amateurs. So this isn't a case of AI stealing mathematicians proofs, or doing something superhuman, its a case of democratization. I am concerned about how AI is affecting math and how the AI companies are behaving, but this isn't the case to be worried about. The calculations for proving this arrangement optimal will always be too big to be checked by hand. However, I'm hoping to produce some nice visualizations of the packing LP or core overlap that rejects each configuration
It is not as arbitrary or ugly as it may seem at first - see the image here and the explanation: <a href="https://x.com/davidmbudden/status/2107646435659481548" rel="nofollow">https://x.com/davidmbudden/status/2107646435659481548</a>
fwiw The prime site for square in square packing is at <a href="https://kingbird.myphotos.cc/packing/squares_in_squares.html" rel="nofollow">https://kingbird.myphotos.cc/packing/squares_in_squares.html</a><p>The triangular view is most interesting. And a 20 minute video on this view is at
<a href="https://youtu.be/uL5wuiy34rs" rel="nofollow">https://youtu.be/uL5wuiy34rs</a>
Can someone explain why 83 and 87 can't get any smaller?
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.
It is possible they can; it’s not yet proven that the listed packings for 83 and 87 are optimal.
Which of the blocks do you think you could move to shrink the solution? Or are you thinking of a completely different arrangement?
They got updated to be smaller this year, so maybe there's still more gains to be had?
This is cool! Something seems broken in the representation for 1850 and 1765, squares are strangely intersecting.<p>edit: Or maybe something wrong with the way my browser (brave) is rendering it.
More on the 11-squares packing:<p><a href="https://startupfortune.com/ai-models-formally-proved-walter-trumps-1979-square-packing-is-optimal/" rel="nofollow">https://startupfortune.com/ai-models-formally-proved-walter-...</a><p><a href="https://vplevris.medium.com/eleven-squares-one-tiny-gap-and-a-problem-still-unsolved-c6f47b447cfb" rel="nofollow">https://vplevris.medium.com/eleven-squares-one-tiny-gap-and-...</a> (written just days before the new proof!)<p><a href="https://jlevy.github.io/squares/cases/11.html" rel="nofollow">https://jlevy.github.io/squares/cases/11.html</a>
"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. ;)
A list of many square packings, with images:<p><a href="https://jlevy.github.io/squares/" rel="nofollow">https://jlevy.github.io/squares/</a>
The readme has no figures :( describing the packing?
There's a cool figure here: <a href="https://x.com/ojoshe/status/2107590622005924265" rel="nofollow">https://x.com/ojoshe/status/2107590622005924265</a>
I had the same thought! Pics please.<p>EDIT: I found it a few links down. <a href="https://jlevy.github.io/squares/cases/11.html" rel="nofollow">https://jlevy.github.io/squares/cases/11.html</a>
The readme also appears to be entirely LLM-written.<p>I really don't get it. If you think you've done something cool, why wouldn't you want to talk about it in your own words?
Pics or it didn't happen
One of the greatest classes I ever took was "Cybernetics", taught by David Huffman ("the" Huffman). he started out the very first day talking about information theory, into sphere packing, and on to applications of sphere packing to communications.<p>I distinctly remember him concluded with something like "Sphere packing is hard, except in 11 dimenions" or something like that, but when I look at the history, I can't see how he knew that in 1994?
Lot more pics here: <a href="https://jlevy.github.io/squares/" rel="nofollow">https://jlevy.github.io/squares/</a>
It’s unintuitive that a messy configuration of squares can be more optimal than neatly arranging them aligned. And by looking at all the current best solutions it does appear that the neat configurations are usually the best , but not always. How does one explain the messy cases?? Is that about how division can result in irrational numbers, and when the number of optimal squares approach one you end up with the messy squares?
Did an interval-arithmetic branch and bound once, getting the rounding modes right took me weeks.
Wow, I never would have imagined one could prove optimality for that accursed beautiful thing.
So this is a proof that the Walter Trump packing is the optimal packing?
Another interesting video related to these types of problems:
<a href="https://youtu.be/mVH7OPx4QZU" rel="nofollow">https://youtu.be/mVH7OPx4QZU</a>
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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?
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... <i>come on</i>.
The whole point is that you can fit more than by naively stacking them...
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").