Squares in Squares

https://kingbird.myphotos.cc/packing/squares_in_squares.html

Comments

ebolyenMay 26, 2026, 10:29 AM
Very relevant comic:

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

dmdMay 26, 2026, 3:26 PM
See also https://www.reddit.com/media?url=https%3A%2F%2Fi.redd.it%2Fn...
sestepMay 26, 2026, 2:26 AM
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.
yzydserdMay 26, 2026, 2:14 PM
Many squares in circles bests were found this month.

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

amneMay 26, 2026, 7:03 AM
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.

javier_e06May 26, 2026, 1:55 PM
Same here. Non native English speaker. The first rule is that inner squares are of size 1. Always.

Yet, in each example the inner squares shrink. Uh?

It know it was a convention to better show the arrangement, normalizing, yadda yadda.

Yet, Uh?

nuancebydefaultMay 26, 2026, 6:45 PM
The total image size is scaled each time such that each solution takes up the same amount of space. It is easier to browse that way.
zamadatixMay 26, 2026, 7:38 PM
Would you also argue it's odd graphs don't all use the same scale as each other?
charlesriceMay 26, 2026, 10:39 PM
Do you want the graphs with 300 squares to be bigger than your screen, or do you want the graph with 1 square to be 30x30 px for no reason? They're just zoomed.
zamadatixMay 27, 2026, 6:42 AM
That's what I mean, I can't imagine why anyone would argue the same thing of graphs in general so I'm curious what the difference makes it so they find it so odd in this specific case.
gus_massaMay 24, 2026, 9:49 PM
In case you want a challenge, 11 is the smaller that has a solution that has not been proven to be optimal.
bradley13May 26, 2026, 7:44 AM
Some of these are wild. You expect to see something systematic, but they have little gaps between oddly placed squares in the center.
NooneAtAll3May 26, 2026, 4:32 AM
I love 130. "You thought I'm just a 2-wide strip? SIKE, here's 8-degree polynomial!"
onedognightMay 26, 2026, 2:18 PM
Unrelated squares in squares, I think the interjection is PSYCH.
xnxMay 26, 2026, 3:00 PM
Awesome site. Slight peeve that arrangements with a prominent diagonal aren't all oriented in the same direction.
razorbeamzMay 26, 2026, 3:38 AM
Looks like Hiroshi Nagamochi did all the boring work.
npodbielskiMay 26, 2026, 6:52 AM
Why 4 is trivial but 6 had to be proved?
smrqMay 26, 2026, 7:43 AM
The 4 packing takes up 100% of its square; it's trivially optimal. The 6 packing only takes up 2/3 of it, so it's not necessarily obvious that you can't do better.
npodbielskiMay 27, 2026, 6:38 AM
for me it is obvious. If I am reading s=3 as multiplier of side of smaller square to side of bigger square, which means that bigger square side is 3 times the side of smaller one, than it is obvious that it should be poossible to squeeze 9 small squares into bigger square. It is children puzzle after all. What is not obvious here?
throawayontheMay 26, 2026, 1:26 PM
i believe 4, 9, 16, 25 etc are just subdivisions of the unit square (they're perfect squares)

but the text also says "For the $n ≤ 324$ not pictured, the trivial packing (with no tilted squares) is the best known packing." applying 'trivial' to numbers that aren't perfect squares so iunno

zamadatixMay 26, 2026, 7:51 PM
It's two different trivial things. For each, it's just the case which doesn't require doing anything special.

One is trivial proofs, which are where 100% is covered. This doesn't really leave much to prove in terms of whether or not more area can be covered by a different layout.

The other is trivial packings, the very simple type without any tilting or need of gaps between squares. Trivial packings are only sometimes optimal. Of optimal trivial packings, only some can be shown optimal with an aforementioned trivial proof.

matthewfelgateMay 26, 2026, 2:43 PM
Sometimes nature is beautiful and sometimes it isn't.