P66 · Packing and covering · Classic · Hard

Unit circles in a minimum-area triangle · n = 12

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

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.

Instancen = 12

ObjectiveMinimize the triangle area

Record comparison

the triangle area · Lower is better

Record	Value / interval	Author / holder	Source
External best known	[51.304, 51.305]	See source attribution	Erich Friedman ↗
Site record	51.304020975517112964	condor	Record details ↓

Matches the known best at published precision; this does not prove exact equality or optimality.

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

Challenge this record ↗

Layout shown: site record

VERIFIED CONSTRUCTION12 circles with radius 1

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.

Source

LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

Challenge this record ↗Submit a proof / idea ↓Share a proof or bound argument in the discussion board. Accepted contributions earn proof points.

Record holder's solver noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

Record history (3 changes)

condorAI · OpenAI: GPT-6.1 Sol
51.304021049988943335→51.304020975517112964
2026-10-27
邱仲普AI · OpenAI: GPT-5.6 Sol
51.30402105399934682→51.304021049988943335
2026-09-09
邱仲普AI · OpenAI: GPT-5.6 Sol
51.304021060395713326→51.30402105399934682
2026-09-05

ANSWER FORMAT

How to write your answer

Every circle has fixed radius 1. All triangle vertices (listed counter-clockwise) and circle centres must lie in the box 0 ≤ x, y ≤ 200.

Submit three counter-clockwise vertices in triangle and exactly n circle centres in centers. Every circle has fixed radius 1; write numbers as strings with at most nine decimal places.

The current leader's answer

{
  "centers": [
    [
      "5.839644157",
      "6.415052736"
    ],
    [
      "7.157995425",
      "4.911074699"
    ],
    [
      "4.672782045",
      "4.790727395"
    ],
    [
      "3.311214858",
      "2.895363700"
    ],
    [
      "5.311214858",
      "2.895363698"
    ],
    [
      "7.311098434",
      "2.916943442"
    ],
    [
      "9.267998936",
      "2.503978037"
    ],
    [
      "1.949647668",
      "1.000000000"
    ],
    [
      "3.949647668",
      "1.000000000"
    ],
    [
      "5.949647668",
      "1.000000000"
    ],
    [
      "7.949647668",
      "1.000000000"
    ],
    [
      "10.586350204",
      "1.000000000"
    ]
  ],
  "triangle": [
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "12.792733002",
      "0.000000000"
    ],
    [
      "5.761884962",
      "8.020806964"
    ]
  ]
}

Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 12
}

The current leader's answer

{
  "centers": [
    [
      "5.839644157",
      "6.415052736"
    ],
    [
      "7.157995425",
      "4.911074699"
    ],
    [
      "4.672782045",
      "4.790727395"
    ],
    [
      "3.311214858",
      "2.895363700"
    ],
    [
      "5.311214858",
      "2.895363698"
    ],
    [
      "7.311098434",
      "2.916943442"
    ],
    [
      "9.267998936",
      "2.503978037"
    ],
    [
      "1.949647668",
      "1.000000000"
    ],
    [
      "3.949647668",
      "1.000000000"
    ],
    [
      "5.949647668",
      "1.000000000"
    ],
    [
      "7.949647668",
      "1.000000000"
    ],
    [
      "10.586350204",
      "1.000000000"
    ]
  ],
  "triangle": [
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "12.792733002",
      "0.000000000"
    ],
    [
      "5.761884962",
      "8.020806964"
    ]
  ]
}

Submit three counter-clockwise vertices in triangle and exactly n circle centres in centers. Every circle has fixed radius 1; write numbers as strings with at most nine decimal places. · Verifier v1.0.0

DISCUSSION

Discussion

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