MinMax Arena
中/EN
P91 · Packing and covering · Classic

L's in a square

VERIFIED CONSTRUCTIONVERIFIED CONSTRUCTION
n = 6Current record · open

Pack n L-trominoes (three unit cells) into a square. Contact is allowed; interiors cannot overlap. Minimise the container square side.

Formal definition

  • ContainerThe editor and certificate normalize to the unit square [0,1]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,quarter},…]}, with x, y and turn as decimal strings of at most nine places. radius is the piece's short side (the leg, the domino's short side, the cell side); the container side is 1/radius. x,y is the centre of the piece's bounding box; quarter is an integer 0–3, an exact number of right angles applied about the centre before the rational 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 square side = 1/r (r is the piece's short side inside the unit square), smaller is better. Pages, leaderboards and literature records share this unit; nothing is converted by hand.

Getting a feel for it

Source and certificate

External targets are converted with their printed precision; decimal coordinate certificates are separate. Reconstruction loss never lowers the literature target, and a source construction is not automatically an optimality proof.

Source
ONE LEADERBOARD PER n

Current best solutions by n

Each n is an independent record with a page of its own. Open any of them to inspect the current construction, then challenge it.

Discussion (0) ↓
n1
CURRENT RECORD2Source value unresolved
Hard
Answer sourceTrivial.
Solution methodPublished reference construction
n2
CURRENT RECORD3.000000004Source value unresolved
Hard
Answer sourceTrivial.
Solution methodPublished reference construction
n3
CURRENT RECORD3.621320382best known 3.621
Matches the best known
Answer sourceSerhiy Grabarchuk
Solution methodPublished reference construction
n4
CURRENT RECORD4Source value unresolved
Hard
Answer sourceTrivial.
Solution methodPublished reference construction
n5
CURRENT RECORD4Source value unresolved
Hard
Answer sourceTrivial.
Solution methodPublished reference construction
n6
CURRENT RECORD4.949747565best known 4.949
Matches the best known
Answer sourceErich Friedman
Solution methodPublished reference construction
n7
CURRENT RECORD5Source value unresolved
Hard
Answer sourceTrivial.
Solution methodPublished reference construction
n8
CURRENT RECORD5Source value unresolved
Hard
Answer sourceTrivial.
Solution methodPublished reference construction
n9
CURRENT RECORD5.819020028best known 5.81902
Matches the best known
Answer sourceAapo Lipponen
Solution methodPublished reference construction
n10
CURRENT RECORD6.000000025Source value unresolved
Hard
Answer sourceTrivial.
Solution methodPublished reference construction
n11
CURRENT RECORD6.000000025Source value unresolved
Hard
Answer sourceTrivial.
Solution methodPublished reference construction
n12
CURRENT RECORD6.000000025Source value unresolved
Hard
Answer sourceTrivial.
Solution methodPublished reference construction
n13
CURRENT RECORD6.828427284best known 6.828
Matches the best known
Answer sourceMaurizio Morandi
Solution methodPublished reference construction
n14
CURRENT RECORD7.000000043Source value unresolved
Hard
Answer sourceTrivial.
Solution methodPublished reference construction
n15
CURRENT RECORD7.000000043Source value unresolved
Hard
Answer sourceTrivial.
Solution methodPublished reference construction
n16
CURRENT RECORD7.000000043Source value unresolved
Hard
Answer sourceTrivial.
Solution methodPublished reference construction
n17
CURRENT RECORD7.747547168best known 7.747
Matches the best known
Answer sourceMaurizio Morandi
Solution methodPublished reference construction
n18
CURRENT RECORD7.914213785best known 7.914
Matches the best known
Answer sourceMaurizio Morandi
Solution methodPublished reference construction
n19
CURRENT RECORD8Source value unresolved
Hard
Answer sourceTrivial.
Solution methodPublished reference construction
n20
CURRENT RECORD8Source value unresolved
Hard
Answer sourceTrivial.
Solution methodPublished reference construction
n21
CURRENT RECORD8Source value unresolved
Hard
Answer sourceTrivial.
Solution methodPublished reference construction
n22
CURRENT RECORD9.00000001best known 8.656
Hard
Record holderFounding benchmark
Solution methodHuman
n23
CURRENT RECORD8.914213814best known 8.914
Matches the best known
Answer sourceMaurizio Morandi
Solution methodPublished reference construction
n24
CURRENT RECORD9.00000001Source value unresolved
Hard
Answer sourceTrivial.
Solution methodPublished reference construction
n25
CURRENT RECORD9.00000001Source value unresolved
Hard
Answer sourceTrivial.
Solution methodPublished reference construction
n26
CURRENT RECORD9.00000001Source value unresolved
Hard
Answer sourceTrivial.
Solution methodPublished reference construction
n27
CURRENT RECORD9.00000001Source value unresolved
Hard
Answer sourceTrivial.
Solution methodPublished reference construction
n28
CURRENT RECORD10best known 9.828
Hard
Record holderFounding benchmark
Solution methodHuman

Data and citation

Every sub-problem in this family — authoritative scores, proof status, coordinates and sources — lives at the stable address below, published under CC BY 4.0. Scores move as records fall, so cite the generatedAt timestamp the file carries.

GET https://minmaxarena.com/data/ells-in-square.json

Cite the frozen 2026-08 edition: records move, a frozen edition never does, so the citation is still checkable years later.

GET https://minmaxarena.com/data/editions/2026-08/ells-in-square.json
BibTeX (click to copy)
@misc{minmaxarena-ells-in-square-2026-08,
  title  = {{L's in a square} (P91)},
  author = {{MinMax Arena}},
  year   = {2026},
  note   = {Machine-verified records, 2026-08 edition},
  url    = {https://minmaxarena.com/data/editions/2026-08/ells-in-square.json},
  license = {CC BY 4.0}
}
DISCUSSION

Discussion

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