Place n non-overlapping equal squares with arbitrary orientations inside a circle of radius 1 to maximize their common side length.
Formal definition
ContainerA circle of radius 1 centered at (1, 1), with both coordinates in [0, 2]
SubmissionEach square is given as {cx, cy, ux, uy}: center (cx, cy) and one half-edge vector (ux, uy), with the orthogonal half-edge fixed as (−uy, ux)
ConstraintsEach square may rotate freely and must lie wholly inside the circle; no two interiors overlap, and boundary contact is allowed. The public metric is the minimum side length across all squares
ObjectiveMaximize the common side length (the verifier compares squared side lengths using exact arithmetic internally)
Getting a feel for it
Rotational freedom against a circular boundary
Because straight edges cannot conform to a circular arc, only the vertices of a square can touch the boundary. Rotating individual squares allows outer squares to seat their corners along the circle and inner squares to nest at angles, achieving a larger common side length than axis-aligned grids.
Proven optima and open frontier
Optimality is proven for n = 3–7: n = 4 by the International Math Summer Camp (2026), and n = 3, 5, 6, 7 by The-Anh Vu-Le's September 2026 Lean 4 formalization. Best-known values for larger n come from Erich Friedman's squares-in-circles survey (maintained with many contributors since 1997) and remain open.
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/tilted-squares-in-circle.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/tilted-squares-in-circle.json
BibTeX (click to copy)
@misc{minmaxarena-tilted-squares-in-circle-2026-10,
title = {{Tilted equal squares in a circle} (P05)},
author = {{MinMax Arena}},
year = {2026},
note = {Machine-verified records, 2026-10 edition},
url = {https://minmaxarena.com/data/editions/2026-10/tilted-squares-in-circle.json},
license = {CC BY 4.0}
}
DISCUSSION
Discussion
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