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P84 · Packing and covering · Classic · Hard

Equilateral triangles in a disc · n = 10

Pack n unit-side equilateral triangles into a disc. Contact is allowed; interiors cannot overlap. Minimise the container circle radius.

Instancen = 10
ObjectiveMinimize container circle radius
Best known, unproven1.38468Source-precision interval: 1.38468–1.38469 (values inside match)exhibited site certificate 1.384681284Compiled by Erich Friedman; Found by Jonathan Viquerat in June 2026. Source expression: r = 1.38468+.

Formal definition

  • ContainerThe editor and certificate normalize to the closed unit disc centred at (1,1). This is a coordinate convention only; scores use the literature's unit-piece scale. Boundary contact is allowed.
  • SubmissionSubmit {radius, placements:[{x,y,turn},…]}, with decimal strings of at most nine places. x,y are centres; circles require turn=0, polygons use 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 circle radius = 1/(2 sin(pi/3) r), smaller is better. Pages, leaderboards and literature share these units; conversion is automatic.
Open the full editor ↗
VERIFIED CONSTRUCTIONVERIFIED CONSTRUCTION

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
The record-holding arrangement for Equilateral triangles in a disc n = 10, 1.384681284
Current leader

1.384681284

container circle radius

Matches the best known
Answer sourceJonathan Viquerat
Solution methodPublished reference construction
Challenge this record ↗Submit a proof / idea ↓Share a proof or idea in the discussion. Accepted contributions can earn proof points.
Record holder's solver noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

ANSWER FORMAT

How to write your answer

the closed unit disc centred at (1,1)

Submit {radius, placements:[{x,y,turn},…]}, with decimal strings of at most nine places. x,y are centres; circles require turn=0, polygons use turn=tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4.

The current leader's answer

{
  "placements": [
    {
      "turn": "-0.546042338",
      "x": "1.081810374",
      "y": "1.591446248"
    },
    {
      "turn": "-0.000283421",
      "x": "1.515778700",
      "y": "1.360801619"
    },
    {
      "turn": "-0.546042338",
      "x": "0.467828862",
      "y": "1.336153768"
    },
    {
      "turn": "0.023803648",
      "x": "0.890749787",
      "y": "1.220841854"
    },
    {
      "turn": "-0.577728226",
      "x": "1.724051659",
      "y": "0.999589577"
    },
    {
      "turn": "0.023803649",
      "x": "0.276768276",
      "y": "0.965549371"
    },
    {
      "turn": "0.576972436",
      "x": "1.098527817",
      "y": "0.839555136"
    },
    {
      "turn": "-0.000283421",
      "x": "1.515369333",
      "y": "0.638613882"
    },
    {
      "turn": "0.424896039",
      "x": "0.497403236",
      "y": "0.478801894"
    },
    {
      "turn": "-0.122422336",
      "x": "0.902045122",
      "y": "0.378220041"
    }
  ],
  "radius": "0.416955350"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 10,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "-0.546042338",
      "x": "1.081810374",
      "y": "1.591446248"
    },
    {
      "turn": "-0.000283421",
      "x": "1.515778700",
      "y": "1.360801619"
    },
    {
      "turn": "-0.546042338",
      "x": "0.467828862",
      "y": "1.336153768"
    },
    {
      "turn": "0.023803648",
      "x": "0.890749787",
      "y": "1.220841854"
    },
    {
      "turn": "-0.577728226",
      "x": "1.724051659",
      "y": "0.999589577"
    },
    {
      "turn": "0.023803649",
      "x": "0.276768276",
      "y": "0.965549371"
    },
    {
      "turn": "0.576972436",
      "x": "1.098527817",
      "y": "0.839555136"
    },
    {
      "turn": "-0.000283421",
      "x": "1.515369333",
      "y": "0.638613882"
    },
    {
      "turn": "0.424896039",
      "x": "0.497403236",
      "y": "0.478801894"
    },
    {
      "turn": "-0.122422336",
      "x": "0.902045122",
      "y": "0.378220041"
    }
  ],
  "radius": "0.416955350"
}

Submit {radius, placements:[{x,y,turn},…]}, with decimal strings of at most nine places. x,y are centres; circles require turn=0, polygons use turn=tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4. · Verifier v1.0.0

DISCUSSION

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