Squares in an equilateral triangle · n = 39
Pack n unit-side squares into a equilateral triangle. Contact is allowed; interiors cannot overlap. Minimise the container triangle side.
Instancen = 39
ObjectiveMinimize container triangle side
Best known, unproven10.30940Source-precision interval: 10.3094–10.30941 (values inside match)exhibited site certificate 10.309401268Compiled by Erich Friedman; Found by Bhavithran Ananthan in July 2026. Source expression: s = 8 + 4 / √3 = 10.30940+.
Formal definition
- ContainerThe editor and certificate normalize to the equilateral triangle with vertices (0,0), (1,0), (1/2,√3/2). 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.
- Objectivecontainer triangle side = 1/(2 sin(pi/4) r), smaller is better. Pages, leaderboards and literature share these units; conversion is automatic.
