The Squares Project

Work on the square packing problem has exploded in the summer of 2026 thanks to AI-powered research efforts. This Squares Project was begun by Joshua Levy in August 2026 with some initial explorations that obtained new lower bounds for $n=11,17,18,19,20$$n=11,17,18,19,20$ and other low values.

Now several others have obtained results building on this work, including:

• Kleddamag’s certified lower bound of 31/8 for eleven squares.
• Mannaseh Ahmed’s proof of optimality for eleven squares.
• Independently, Evan Daniel has proved the optimality of an infinite family, s(k2−3)=k$s(k^2-3)=k$ for every $k\ge 6$$k\ge 6$, along with exact values at 21, 32 and 45 squares.

This project now independently tabulates all known new results and does AI-assisted verification of the proofs and certificates behind them, to encourage open collaboration on open questions and formalizations of current proofs.

Contribute Your Results!

If you have new results or know of newer results, please file an issue to report them, and we will gladly incorporate them and cite your work.

We also have a group chat. Contact ojoshe if you wish to join.

Squares Project Documentation

The code, the certificates, the literature archive and the documents that record all of this live in the Squares Project’s repository.