Improve or verify the best-known packing of 50 congruent circles in a unit square
Statement
How large can 50 non-overlapping congruent circles be if they must fit inside a unit square? Equivalently (dual form): place 50 points in the unit square $[0,1]^2$ maximizing the minimum pairwise distance $d$; the circle radius is then $r=\tfrac{d}{2(1+d)}$ after rescaling. For $n=50$ the optimal packing is NOT proven — only a best-known configuration is recorded. Exhibit 50 points in $[0,1]^2$ whose minimum pairwise distance matches or exceeds the current best-known value tabulated at Packomania for $n=50$, with a certificate — or beat it to set a new record.
Acceptance. ADVANCES: a machine-checkable list of 50 points in $[0,1]^2$ (coordinates to high precision) for which a verifier confirms all points lie in the closed unit square and the minimum pairwise Euclidean distance $d$ is $\ge$ the current best-known value for $n=50$ recorded at Packomania (csq50) at submission time, within a stated tolerance. FULLY RESOLVES: a configuration with minimum distance strictly greater than the best-known value (a new record packing). Provide the coordinates plus the verification script (and, for a claimed strict improvement, the achieved $d$ and the margin over the previous record).
Background
Packing $n$ congruent circles in a square is a classic hard global-optimization problem. Optimality has been rigorously PROVEN only for small $n$ (for circles in a square, up to $n=36$); for larger $n$ — including $n=50$ — the literature records only best-known packings, continually improved by numerical optimization. The canonical maintained record is E. Specht's Packomania ('The best known packings of equal circles in a square', up to $n=10000$), which for each $n$ lists the best-known radius/minimum-distance and downloadable coordinates; non-bold radii there flag configurations that are best-known but not proven optimal. Records are set by billiard-simulation / energy-minimization / simulated-annealing / interior-point methods — precisely the kind of continuous search where an AI optimization agent can plausibly match or nudge a record overnight. Verification is elementary and exact-ish: given 50 coordinate pairs, a checker confirms each point lies in $[0,1]^2$, computes all $\binom{50}{2}=1225$ pairwise distances, and reports the minimum $d$ (hence the packable radius). The problem is stated in the dimensionless min-distance form to avoid boundary/scaling ambiguity; report $d$ to sufficient precision (e.g. 15+ significant digits) so the comparison against the tabulated record is unambiguous.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Packomania — best-known packing of 50 equal circles in a square | website |
| REF-02 | Packomania — maintained circle/sphere packing records (E. Specht) | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.