Packing equal circles in a square

What is the largest radius that $N$ equal circles can have inside a unit square without overlapping? Optimal packings are proved only for $N \le 30$ and $N = 36$ ; Eckard Specht's Packomania lists the best known packings up to $N = 10000$ , and any improvement can be checked exactly in milliseconds.

Problem

For $N \ge 2$ , let $r_N$ be the largest radius such that $N$ circles of radius $r_N$ fit in the unit square with disjoint interiors. Equivalently, place $N$ points in the unit square so as to maximize the smallest distance $m_N$ between two of them; the two problems are related by $r_N = m_N / (2(m_N + 1))$ .

Task: find packings that beat the records in Packomania's table, or prove that a record is optimal. The smallest case without a proof is $N = 31$ .

Background

The problem is a classical test bed for global optimization and computational geometry, and dense packings for larger $N$ often follow repeated patterns (Graham and Lubachevsky). Proofs of optimality for small $N$ came first from geometric arguments and later from computer-aided methods, culminating in interval-arithmetic proofs that settled 28, 29 and 30 circles. Packomania, maintained by Eckard Specht, collects the records with coordinates, densities, contact graphs and references, alongside many other containers.

What is known

Proved optimal (bold in Packomania's table): all $N \le 30$ and $N = 36$ . Proofs include Kirchner and Wengerodt (1987) for 36 circles, computer-aided proofs by Nurmela and Östergård (1999), Markót (2004) for 28 circles, and Markót and Csendes (2005), who used verified interval methods for 28 to 30 circles.
Smallest open case: $N = 31$ , with best known radius $0.089338333351$ (distance between centres $0.217547291619$ in the point formulation).
Records: best known packings for every $N$ up to 10000, contributed by many people; the circles-in-a-square table was last updated on 10 September 2026.
Submissions: in October 2026 the site asked contributors not to submit further entries that month, because more than 5,000 new candidates were waiting in the queue. New packings can still be found and checked now, and submitted once the queue reopens.

What would count as progress

A packing with a larger radius than the record for some $N$ . Checking it means verifying containment and all $\binom{N}{2}$ pairwise distances; with interval or exact arithmetic this takes milliseconds even for large $N$ , and the site's format specifies how to submit coordinates.
A proof of optimality for $N = 31$ or another small open case, for example with the interval branch-and-bound techniques used for 28 to 30 circles, published with code that can be rerun.
The same questions for Packomania's other containers, such as circles in a circle, which have their own tables.

References

K. J. Nurmela and P. R. J. Östergård, "More optimal packings of equal circles in a square", Discrete & Computational Geometry 22 (1999), 439–457. doi:10.1007/pl00009472
M. Cs. Markót, "Optimal packing of 28 equal circles in a unit square: the first reliable solution", Numerical Algorithms 37 (2004), 253–261. doi:10.1023/b:numa.0000049472.75023.0a
M. Cs. Markót and T. Csendes, "A new verified optimization technique for the packing circles in a unit square problems", SIAM Journal on Optimization 16 (2005), 193–219. doi:10.1137/s1052623403425617
P. G. Szabó and E. Specht, "Packing up to 200 equal circles in a square", in Models and Algorithms for Global Optimization, Springer (2007), 141–156. doi:10.1007/978-0-387-36721-7_9

Source

Imported from Packomania, circles in a square, curated by Eckard Specht. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.