P66 · Packing and covering · Classic

Unit circles in a minimum-area triangle

Also known ascircles in arbitrary trianglesunit circles in a minimum-area trianglecircle packing in a triangle of variable shape

VERIFIED CONSTRUCTION10 circles with radius 1

n = 10Current record · open

Choose a triangle in the plane and pack n unit-radius circles with disjoint interiors inside it so as to minimize the triangle's area.

Formal definition

ContainerA non-degenerate triangle whose shape is not fixed in advance; all vertices and circle centres must lie in [0, 200]²
SubmissionThree triangle vertices in counter-clockwise order and exactly n circle centres
CirclesEvery circle has fixed radius 1; circle interiors are pairwise disjoint and tangency is allowed
ContainmentEvery circle lies wholly inside the triangle and may be tangent to an edge
ObjectiveMinimize the area of the triangle

Getting a feel for it

Coupled boundary and packing

Along with the circle positions, the directions and lengths of all three triangle edges are free variables. Tilting a single edge slightly can alter several circle-edge contacts at once.

Rigid contact graphs

Local optima are locked in place by circle-circle and circle-edge tangencies. Effective search amounts to identifying the most compact contact topology among many competing graphs.

Known constructions and open cases

Friedman's public table records the best-known constructions up to n = 50. Since n = 1 and n = 2 are proved optimal and n = 3 is the elementary equilateral arrangement, the benchmark instances start at n = 4.

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 (0) ↓

CURRENT RECORD19.3923048486324840815best known 9 + 6√3

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

CURRENT RECORD24.124355653180826755best known 12 + 7√3

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

CURRENT RECORD24.124355659716054925best known 12 + 7√3

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Terra

CURRENT RECORD31.2121776589315350325best known [31.212, 31.213]

Matches the best known

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Terra

CURRENT RECORD35.93642283165247292best known [35.936, 35.937]

Matches the best known

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Terra

CURRENT RECORD38.784609693336059739best known 18 + 12√3

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Terra

CURRENT RECORD38.784609697279413042best known 18 + 12√3

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Terra

CURRENT RECORD47.003545251932505066best known [47.003, 47.004]

Matches the best known

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

CURRENT RECORD51.304020975517112964best known [51.304, 51.305]

Matches the best known

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

CURRENT RECORD54.8949652768913927675best known [54.894, 54.895]

Matches the best known

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

CURRENT RECORD56.90896535489462367best known 24 + 19√3

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Terra

CURRENT RECORD56.908965360132545964best known 24 + 19√3

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Terra

CURRENT RECORD66.033321017593110428best known 21 + 26√3

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

CURRENT RECORD70.2750675121253867105best known [70.275, 70.276]

Matches the best known

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

CURRENT RECORD74.8033529496800224215best known [74.803, 74.804]

Matches the best known

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

CURRENT RECORD77.34617465654030966best known [77.346, 77.347]

Matches the best known

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

CURRENT RECORD78.497422624711541437best known 30 + 28√3

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Terra

CURRENT RECORD78.497422628688281192best known 30 + 28√3

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Terra

CURRENT RECORD89.1277441690301888935best known [89.127, 89.128]

Matches the best known

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

CURRENT RECORD92.741380391028567576best known [92.74138, 92.74139]

Matches the best known

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

CURRENT RECORD96.497422650875826144best known 48 + 28√3

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

CURRENT RECORD99.746133953452588696best known 27 + 42√3

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

CURRENT RECORD102.7048681448774838585best known [102.704868097, 102.70488]

Matches the best known

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Terra

CURRENT RECORD103.54998152436426135best known 36 + 39√3

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Terra

CURRENT RECORD103.549981514242745283best known 36 + 39√3

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Terra

CURRENT RECORD115.292242985694923227best known [115.29224, 115.29225]

Matches the best known

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

CURRENT RECORD118.848622274212988655best known [118.84862, 118.84863]

Matches the best known

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

CURRENT RECORD122.58804469790485588best known [122.58804, 122.58805]

Matches the best known

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

CURRENT RECORD127.074904016320079535best known [127.07490, 127.07491]

Matches the best known

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Terra

CURRENT RECORD129.206348134521487921best known [129.206348075, 129.20636]

Matches the best known

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

CURRENT RECORD131.585922831270990232best known [131.58592, 131.58593]

Matches the best known

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

CURRENT RECORD132.0666420093831393505best known 42 + 52√3

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Terra

CURRENT RECORD132.066641995316021286best known 42 + 52√3

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

CURRENT RECORD144.877234995027470208best known [144.87723, 144.87724)

Matches the best known

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

CURRENT RECORD148.37140971980877121best known [148.371409637, 148.37142]

Matches the best known

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

CURRENT RECORD151.830724821787259128best known [151.830724748, 151.83074]

Matches the best known

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

CURRENT RECORD155.1384388208496376best known 72 + 48√3

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

CURRENT RECORD159.440748796630442832best known [160.08329, 160.08330)

Numerically ahead (under review)

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

CURRENT RECORD161.88709138917473461best known [161.88709, 161.88710]

Matches the best known

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

CURRENT RECORD163.7193273675825288605best known [163.719327302, 163.71934]

Matches the best known

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

CURRENT RECORD164.047404155357160294best known 48 + 67√3

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Terra

CURRENT RECORD164.047404138929743008best known 48 + 67√3

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Terra

CURRENT RECORD177.887369133417315928best known [177.887369044, 177.88738]

Matches the best known

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

CURRENT RECORD181.356482341027393544best known [181.35648, 181.35649)

Matches the best known

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

CURRENT RECORD184.6142485229754459375best known [184.75052, 184.75053]

Numerically ahead (under review)

Record holderzzzcy

Solution methodAI · OpenAI: GPT-6 Astra

CURRENT RECORD187.956369497045302558best known 84 + 61√3

Numerically ahead (under review)

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

CURRENT RECORD193.318101562948860608best known [194.49896, 194.49897]

Numerically ahead (under review)

Record holderzzzcy

Solution methodAI · OpenAI: GPT-6 Astra

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/unit-circles-in-minimum-area-triangle.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/unit-circles-in-minimum-area-triangle.json

BibTeX (click to copy)

@misc{minmaxarena-unit-circles-in-minimum-area-triangle-2026-10,
  title  = {{Unit circles in a minimum-area triangle} (P66)},
  author = {{MinMax Arena}},
  year   = {2026},
  note   = {Machine-verified records, 2026-10 edition},
  url    = {https://minmaxarena.com/data/editions/2026-10/unit-circles-in-minimum-area-triangle.json},
  license = {CC BY 4.0}
}

DISCUSSION

Discussion

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