Equal circles in regular polygons · m = 8 · n = 40
Pack n equal circles into the regular m-gon with centre (1,1), circumradius 1, and first vertex (2,1) so as to maximise their common radius.
Instancem = 8 · n = 40
ObjectiveMaximize circle radius
Record comparison
circle radius · Higher is better| Record | Value / interval | Author / holder | Source |
|---|---|---|---|
| External best known | [0.132682454148, 0.13268245415] | Paolo Amore | Packomania ↗ |
| Site record | No player record yet | — | — |
Formal definition
- ContainerThe editor and certificate normalise the container to the exact regular m-gon with centre (1,1), circumradius 1, and first vertex (2,1), while public scores follow the literature convention. Boundary contact is allowed.
- SubmissionSubmit {radius, placements:[{x,y,turn},…]}, with each value written as a decimal string of at most nine places. Here x, y are centre coordinates; circles require turn = 0 and polygons use turn = tan(θ/2). Coordinates and turn must lie in [−4, 4], with 0 < radius ≤ 4.
- ConstraintsExactly n congruent pieces lie wholly inside the container with pairwise disjoint interiors. Polygons may rotate independently by arbitrary angles.
- PrecisionInput decimals are parsed as exact rationals and regular-polygon vertices are defined algebraically, with zero floating-point tolerance in feasibility checks.
- ObjectiveMaximise the circle radius r (with container circumradius fixed at 1).
