I guess when my parcel arrives all crumbled it's because it was along other 23 parcels instead of 22 and the courier knows his optimal square packing solutions.
replyThanks for posting this, I remember seeing this some time ago but I had lost the link.
replyCan someone explain why 83 and 87 can't get any smaller?
replyBecause 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.
replyIt is possible they can; it’s not yet proven that the listed packings for 83 and 87 are optimal.
replyWhich of the blocks do you think you could move to shrink the solution? Or are you thinking of a completely different arrangement?
replyThey got updated to be smaller this year, so maybe there's still more gains to be had?
reply"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. ;)
replyMore on the 11-squares packing:
replyhttps://startupfortune.com/ai-models-formally-proved-walter-...
https://vplevris.medium.com/eleven-squares-one-tiny-gap-and-... (written just days before the new proof!)
This is cool! Something seems broken in the representation for 1850 and 1765, squares are strangely intersecting.
replyedit: Or maybe something wrong with the way my browser (brave) is rendering it.
I am not a mathematician. Why can a square of s = N not usually fit N * N unit squares? For example S = 4, one would naively (I guess) think it could fit 16 unit squares (4 * 4), but if I’m reading the above correct the actual solution is 15 (with what looks like a unit-square-sized space left over). Same for S = 5, and S = 6, but not for S = 2, which fits the expected 4 unit squares.
replyThey've just excluded square numbers because (as you noted) the solution is obvious. Except 4, for some reason.
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