Equal circles in regular polygons · m = 5 · n = 31
Pack n equal circles into the true regular m-gon with centre (1,1), circumradius 1 and first vertex (2,1), maximising their radius.
Instancem = 5 · n = 31
ObjectiveMaximize circle radius
Best known, unproven0.138616362210Source-precision interval: 0.138616362209–0.138616362211 (values inside match)exhibited site certificate 0.138616361Packomania (compiled by Eckard Specht), m=5, n=31. Eckard Specht, program cpt , 2023. Coordinates imported from the public archive, normalized, decimalized and exactly verified by MinMax Arena. The web-table target and grid certificate are separate.
Formal definition
- ContainerThe editor and certificate normalize to the true regular m-gon with centre (1,1), circumradius 1 and first vertex (2,1). This is a coordinate convention only; scores use the literature's unit-piece scale. Boundary contact is allowed.
- SubmissionSubmit {radius, placements:[{x,y,turn},…]}, with decimal strings of at most nine places. x,y are centres; circles require turn=0, polygons use turn=tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4.
- ConstraintsExactly n congruent shapes lie wholly in the container with pairwise disjoint interiors. Polygon rotations are independent, not limited to quarter turns.
- PrecisionInput decimals are exact rationals; regular-polygon vertices are algebraic, not rounded templates. Feasibility has no floating-point tolerance. A finite-decimal certificate is not a continuous optimality proof.
- ObjectiveCircle radius r, larger is better; the container circumradius is 1.
