Squares in Squares

Very relevant comic:

https://thejenkinscomic.wordpress.com/2024/12/01/brady-bunch...

The triangular table view is fascinating. It looks like the periodic table. I wonder if there are number-theoretic lemmas (or at least conjectures?) about what "family" the optimal packing for a given number falls into (like diamond, diagonal strip, two blobs, etc). I didn't see anything when skimming the survey paper linked at the bottom of the site, but I'm sure there's a lot more literature here.

Many squares in circles bests were found this month.

https://erich-friedman.github.io/packing/squincir/

Awesome site. Slight peeve that arrangements with a prominent diagonal aren't all oriented in the same direction.

if, like me, you're a non-native english and speaker don't immediately understand what this is about: the page shows for each `n` what's the minimum `s` such that `n` squares with side of length 1 fit in a square with side of length `s`.

what I'm curious about though is what a proof for something like this looks like. and why does it need a proof? not to mention the randomness of some of the `n`s. Math is most of the time beatiful and whenever I see something like `n=11` I think "it looks wrong so it must be wrong" yet it has a proof.

Some of these are wild. You expect to see something systematic, but they have little gaps between oddly placed squares in the center.

In case you want a challenge, 11 is the smaller that has a solution that has not been proven to be optimal.

I love 130. "You thought I'm just a 2-wide strip? SIKE, here's 8-degree polynomial!"

Sometimes nature is beautiful and sometimes it isn't.

Looks like Hiroshi Nagamochi did all the boring work.

Why 4 is trivial but 6 had to be proved?