Pack n unit squares into an equilateral triangle container with boundary contact allowed and disjoint interiors, minimising the side length of the container triangle.
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.
Getting a feel for it
Literature targets and coordinate certificates
Literature targets are converted at their published precision, while nine-decimal coordinate certificates are stored and displayed alongside the source values.
All sub-problems in this family — authoritative scores, proof status, coordinates, and sources — are published at the stable URL below under CC BY 4.0. Because scores update as records fall, please cite the file's generatedAt timestamp.
GET https://minmaxarena.com/data/squares-in-triangle.json
Cite the frozen 2026-10 edition: live records change, whereas a frozen edition stays fixed and reproducible.
GET https://minmaxarena.com/data/editions/2026-10/squares-in-triangle.json
BibTeX (click to copy)
@misc{minmaxarena-squares-in-triangle-2026-10,
title = {{Squares in an equilateral triangle} (P85)},
author = {{MinMax Arena}},
year = {2026},
note = {Machine-verified records, 2026-10 edition},
url = {https://minmaxarena.com/data/editions/2026-10/squares-in-triangle.json},
license = {CC BY 4.0}
}
DISCUSSION
Discussion
Discuss strategies, share construction methods, or ask questions. Any signed-in user can post under their public byline and unique #UID. New posts appear after moderation.
Grouped by player and sub-problem, showing the latest note first. Earlier versions and replies are preserved; click any note to read or reply. Notes can be published alongside record-breaking submissions.