Tilted equal squares in the unit square
Also known assquares in squarespacking unit squares in a squaretilted square packing
Pack n equal squares inside the unit square, each free to tilt at any angle without overlapping, so that their common side length is as large as possible.
Formal definition
- ContainerThe unit square [0, 1] × [0, 1], with lower-left corner (0, 0) and upper-right corner (1, 1)
- 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 unit square; 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
How tilted squares save space
When n is not a perfect square, axis-aligned grids leave unused strips of space. Allowing rotations lets subsets of squares tilt into those gaps; for example, the optimal configuration for n = 5 places four axis-aligned squares in the corners and one 45° tilted square in the center.
Proven optima and open frontier
This is the classic square-in-square packing problem (often formulated via the minimum container side s(n)): optimality is proven for n = 3, 4, 6..9, 14..16, and 25 (Göbel, Kearney–Shiu, and others), while the remaining cases are open. Erich Friedman's squares survey tracks the best-known constructions.
SourceCurrent best solutions by n
Each parameter n is an independently scored sub-problem. Select any card to inspect the current record construction or submit a stronger solution.
Discussion (3) ↓Data and citation
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-square.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-square.json
BibTeX (click to copy)
@misc{minmaxarena-tilted-squares-in-square-2026-10,
title = {{Tilted equal squares in the unit square} (P18)},
author = {{MinMax Arena}},
year = {2026},
note = {Machine-verified records, 2026-10 edition},
url = {https://minmaxarena.com/data/editions/2026-10/tilted-squares-in-square.json},
license = {CC BY 4.0}
}Discussion
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