P15 · Packing and covering · Solved · Classic

Spreading points in the unit square

1y0

123456

0x1

VERIFIED CONSTRUCTIONthe closest pair

n = 6Current record · open

Place n points in the unit square to maximize the minimum Euclidean distance between any two points.

Formal definition

ContainerThe unit square [0, 1] × [0, 1], with lower-left corner (0, 0) and upper-right corner (1, 1)
SubmissionExactly n distinct points
ConstraintsEvery point lies inside the unit square or on its boundary
Scoreδ(P) is the minimum Euclidean distance over all point pairs (the verifier compares δ(P)² using exact arithmetic internally) $δ (P) = \min_{1 \leq i < j \leq n} ‖ p_{i} - p_{j} ‖$
ObjectiveMaximize the minimum pairwise distance δ(P)

Getting a feel for it

Duality with equal-circle packing in a square

Placing n points in the unit square with minimum pairwise distance d is equivalent to packing n non-overlapping circles of radius d/2 in a square of side 1 + d (or radius r = d / (2 + 2d) in the unit square). At optimality, point pairs at distance d and points on the four edges lock into a rigid contact graph.

Proven optima and open frontier

Through the exact conversion to equal-circle packing in a square (csq), proofs and constructions in the csq literature transfer directly. Elementary partition proofs are included here for n = 2, 4, 5, and 9, while the remaining sizes follow the csq benchmark table.

Source

ONE LEADERBOARD PER n

Current 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) ↓

CURRENT RECORD1.414213562

Optimal

Answer sourceReference answer

Solution methodPublished reference construction

CURRENT RECORD1.03527618proven optimum √6 − √2

Hard

123

Record holderZrchedcxen

Answer sourceReference answer

Solution methodPublished reference construction

CURRENT RECORD0.707106781

Optimal

12345

Answer sourceReference answer

Solution methodPublished reference construction

CURRENT RECORD0.54810912proven optimum √13/6

Hard

123456

Record holderZrchedcxen

Solution methodHuman

CURRENT RECORD0.486793262proven optimum 4 − 2√3

CURRENT RECORD0.516623654proven optimum (√6 − √2)/2

Answer sourceReference answer

Solution methodPublished reference construction

n10

CURRENT RECORD0.360987525proven optimum 0.421279543

CURRENT RECORD0.25proven optimum 0.39820731

Hard

1234567891011

Record holderFounding benchmark

Solution methodHuman

n12

CURRENT RECORD0.25proven optimum 0.388730126

Hard

123456789101112

Record holderFounding benchmark

Solution methodHuman

This problem is frozen

Mathematically equivalent to P01 (equal circles in the unit square) under d = 2r/(1 − 2r); Specht's reference table lists both quantities as columns of the same table. The catalogue retains the circle-packing formulation (P01).

Closed to new records since 2026-08-28. Existing records remain in the history and on their holders' profiles, and this URL stays active; since records can no longer change hands, they no longer count toward contribution points as of 2026-10-04.

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/spread-points-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/spread-points-in-square.json

BibTeX (click to copy)

@misc{minmaxarena-spread-points-in-square-2026-10,
  title  = {{Spreading points in the unit square} (P15)},
  author = {{MinMax Arena}},
  year   = {2026},
  note   = {Machine-verified records, 2026-10 edition},
  url    = {https://minmaxarena.com/data/editions/2026-10/spread-points-in-square.json},
  license = {CC BY 4.0}
}

DISCUSSION

Discussion

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冻结，但贡献值……NUE_13#310 replies31d ago