Submit progress

on Circle packing in a square / rectangle (AlphaEvolve problem 36)

The statement

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.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.