Squares in an equilateral triangle · n = 27
Pack n unit squares into an equilateral triangle container with boundary contact allowed and disjoint interiors, minimising the side length of the container triangle.
Instancen = 27
ObjectiveMinimize container triangle side
Record comparison
container triangle side · Lower is better| Record | Value / interval | Author / holder | Source |
|---|---|---|---|
| External best known | [8.618, 8.619) | Erich Friedman | Erich Friedman ↗ |
| Site record | 8.618802239 | 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 equilateral triangle with vertices (0,0), (1,0), and (1/2,√3/2), 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 triangle side = 1/(2 sin(π/4) r), where smaller is better. Pages, leaderboards, and literature records share this unit with automatic conversion.
