Equilateral triangles in a square · n = 40
Pack n unit-side equilateral triangles into a square. Contact is allowed; interiors cannot overlap. Minimise the container square side.
Instancen = 40
ObjectiveMinimize container square side
Record comparison
container square side · Lower is better| Record | Value / interval | Author / holder | Source |
|---|---|---|---|
| External best known | [4.41057, 4.41058) | 40. Found by Thomas Schadt | Erich Friedman ↗ |
| Site record | 4.410574386 | NUE_13 | Record details ↓ |
Matches the known best at published precision; this does not prove exact equality or optimality.
Formal definition
- ContainerThe editor and certificate normalise the container to the unit square [0,1]², while public scores are reported in the literature's unit-piece scale. 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.
- ObjectiveContainer square side = 1/(2 sin(π/3) r), where smaller is better. Pages, leaderboards, and literature records share this unit with automatic conversion.
