Consecutive dominoes in a square
Place n dominoes, the k-th a k × 2k rectangle (k = 1, …, n), without overlap in a square, moved and turned freely, so as to minimise the square's side.
Formal definition
- ContainerThe closed square [0, side] × [0, side] with its lower-left corner at the origin, in units of the smallest domino's short side; dominoes may touch its sides
- SubmissionThe square's side, side > 0, and the placement of exactly n dominoes: the k-th (k = 1, …, n) is the closed k × 2k rectangle, each used exactly once; every value has at most nine decimal places
- PlacementEach piece may be moved and turned freely: rotated lays it flat (2k × k) or stands it up (k × 2k), then it turns counter-clockwise by θ about its lower-left corner (x, y), with tan(θ/2) = turn. cos θ and sin θ are rational, so every corner is rational and any orientation can be written
- ConstraintsEvery piece's four corners lie in the square; no two pieces' interiors intersect (touching is allowed), decided by the separating-axis test on both pieces' edge normals. Everything is decided exactly in integers, never approximately
- ScoreThe square's side, displayed to at most nine decimal places; smaller is better
- ObjectiveMinimise the square's side over all valid arrangements
Getting a feel for it
The area and the longest piece
The dominoes cover 2(1² + 2² + … + n²) = n(n + 1)(2n + 1)/3 in all, so the side is at least the square root of that; the longest piece also needs a side of at least 2n, which is what decides n ≤ 4. At n = 41 the area bound is about 218.3, and the literature's 220 wastes less than 2% of the square: the large pieces set the frame, the small ones fill the gaps, and almost nothing is left over.
Literature records
Erich Friedman's table Consecutive Dominoes in Squares covers n = 1–41; every side is an integer and every pictured packing is axis-aligned. n = 1–4 are the trivial s = 2n; most of n = 5–24 are Friedman's own, from 1998 to 2014, with n = 20 by Maurizio Morandi in 2014, the one row the table says may not be optimal; n = 25–38 come from Matthew Sun and others in June 2026, with n = 26 improved by Ryan Chi and d/dx in September; and Rowan Jones added n = 39–41 in October 2026. This catalogue cites each row's value and author without reconstructing its configuration; every size starts from the site's ordinary shelf packing, behind the literature value and not the external record.
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 (1) ↓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/consecutive-dominoes-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/consecutive-dominoes-in-square.json
BibTeX (click to copy)
@misc{minmaxarena-consecutive-dominoes-in-square-2026-10,
title = {{Consecutive dominoes in a square} (P95)},
author = {{MinMax Arena}},
year = {2026},
note = {Machine-verified records, 2026-10 edition},
url = {https://minmaxarena.com/data/editions/2026-10/consecutive-dominoes-in-square.json},
license = {CC BY 4.0}
}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.
No solver notes yet.