Dominoes in a square · n = 33
Pack n 1 × 2 dominoes into a square container with boundary contact allowed and disjoint interiors, minimising the side length of the container square.
Instancen = 33
ObjectiveMinimize container square side
Record comparison
container square side · Lower is better| Record | Value / interval | Author / holder | Source |
|---|---|---|---|
| External best known | [8.707106781, 8.70712) | Joshua Wolk | Erich Friedman ↗ |
| Site record | 8.70710679 | 邱仲普 | 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 1 × 2 domino scale. Boundary contact is allowed.
- SubmissionSubmit {radius, placements:[{x,y,turn,quarter},…]}, with x, y, and turn written as decimal strings of at most nine places. Here radius is the short side of each domino inside the unit-square container, so the container side length is 1/radius. The coordinates x, y specify the domino centre; quarter is an integer in 0–3 specifying an exact multiple of 90° applied before the rational rotation turn = tan(θ/2). Coordinates and turn must lie in [−4, 4], with 0 < radius ≤ 4.
- ConstraintsExactly n congruent dominoes lie wholly inside the container with pairwise disjoint interiors. Dominoes may rotate independently by arbitrary angles.
- PrecisionInput decimals are parsed as exact rationals, with zero floating-point tolerance in feasibility checks.
- ObjectiveContainer square side = 1/r (where r is the domino's short side inside the unit square), where smaller is better. Pages, leaderboards, and literature records share this unit with automatic conversion.
