Circle packing in a square / rectangle

The problem

For any $n \geq 1$, let $C(n)$ denote the largest sum $\sum_{i=1}^n r_i$ of radii such that one can place $n$ disjoint open disks of radius $r_1,\dots,r_n$ inside the unit square, and let $C'(n)$ denote the largest sum $\sum_{i=1}^n r_i$ of radii such that one can place $n$ disjoint open disks of radius $r_1,\dots,r_n$ inside a rectangle of perimeter $4$. Establish upper and lower bounds for $C(n)$ and $C'(n)$ that are as strong as possible.

Statement quoted from problem 36 of the AlphaEvolve repository of problems (Google DeepMind, CC BY 4.0).

What AI did

AlphaEvolve improved the best known bound; the record has since been surpassed. The problem remains open. Status in the repository: Former new result, since improved upon. See Section 6.19 in "Mathematical Exploration and Discovery at Scale" paper. Records may have moved since; the repository's notes below list the latest bounds it knows of.

Notes from the repository

Slight improvements to the packings generated by AlphaEvolve have since been constructed by Timo Berthold and by Arnaud Deza.

How it was checked

The results are explicit constructions or programs; the repository's notebooks include the verification code used to score them.

Limitations

An improved bound is progress, not a solution: the exact answer to the problem is still unknown. The authors report that results were better when experts set up and guided the search.

∎