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P84 · Packing and covering · Classic · Hard

Equilateral triangles in a disc · n = 32

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

Instancen = 32
ObjectiveMinimize container circle radius
Best known, unproven2.31295Source-precision interval: 2.31295–2.31296 (values inside match)exhibited site certificate 2.312955248Compiled by Erich Friedman; Found by Thomas Schadt in August 2026. Source expression: r = 2.31295+.

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 = 32, 2.312955248
Current leader

2.312955248

container circle radius

Matches the best known
Answer sourceThomas Schadt
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": "0.577232428",
      "x": "0.997305461",
      "y": "1.776359860"
    },
    {
      "turn": "-0.000088385",
      "x": "1.246921299",
      "y": "1.776315736"
    },
    {
      "turn": "0.535854130",
      "x": "0.621776579",
      "y": "1.762941251"
    },
    {
      "turn": "-0.000088385",
      "x": "0.872459329",
      "y": "1.560208266"
    },
    {
      "turn": "-0.000088385",
      "x": "1.590572587",
      "y": "1.542338704"
    },
    {
      "turn": "0.577232428",
      "x": "1.340956749",
      "y": "1.542382828"
    },
    {
      "turn": "-0.165644408",
      "x": "0.421100098",
      "y": "1.533875848"
    },
    {
      "turn": "-0.000088385",
      "x": "1.216110617",
      "y": "1.326231234"
    },
    {
      "turn": "0.577232428",
      "x": "0.966494779",
      "y": "1.326275359"
    },
    {
      "turn": "0.577232428",
      "x": "1.715342293",
      "y": "1.326142984"
    },
    {
      "turn": "0.375769230",
      "x": "0.349979002",
      "y": "1.265354556"
    },
    {
      "turn": "-0.000088385",
      "x": "0.662593946",
      "y": "1.213543250"
    },
    {
      "turn": "-0.577468122",
      "x": "1.340880323",
      "y": "1.110035514"
    },
    {
      "turn": "-0.000088385",
      "x": "1.590496161",
      "y": "1.109991390"
    },
    {
      "turn": "-0.017019833",
      "x": "0.148946192",
      "y": "1.028977981"
    },
    {
      "turn": "-0.000088385",
      "x": "1.036979490",
      "y": "0.997303406"
    },
    {
      "turn": "0.577232428",
      "x": "0.787363652",
      "y": "0.997347531"
    },
    {
      "turn": "-0.577468122",
      "x": "1.843956888",
      "y": "0.968065138"
    },
    {
      "turn": "0.577232428",
      "x": "0.412919979",
      "y": "0.884752148"
    },
    {
      "turn": "-0.000088385",
      "x": "0.662517520",
      "y": "0.781195936"
    },
    {
      "turn": "0.577232428",
      "x": "1.161749196",
      "y": "0.781107687"
    },
    {
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      "x": "1.411365034",
      "y": "0.781063562"
    },
    {
      "turn": "-0.000088385",
      "x": "0.288073847",
      "y": "0.668600553"
    },
    {
      "turn": "-0.577468122",
      "x": "1.683996049",
      "y": "0.628064821"
    },
    {
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      "x": "1.036903064",
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    },
    {
      "turn": "-0.577468122",
      "x": "0.787287225",
      "y": "0.565000217"
    },
    {
      "turn": "-0.577468122",
      "x": "0.412843553",
      "y": "0.452404834"
    },
    {
      "turn": "0.577232428",
      "x": "1.309537992",
      "y": "0.434095551"
    },
    {
      "turn": "-0.000088385",
      "x": "1.559149917",
      "y": "0.411913226"
    },
    {
      "turn": "-0.000088385",
      "x": "0.662441093",
      "y": "0.348848622"
    },
    {
      "turn": "-0.577468122",
      "x": "0.935076022",
      "y": "0.217988081"
    },
    {
      "turn": "-0.000088385",
      "x": "1.184691860",
      "y": "0.217943956"
    }
  ],
  "radius": "0.249615841"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 32,
  "sides": 5
}

The current leader's answer

{
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      "x": "0.997305461",
      "y": "1.776359860"
    },
    {
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      "y": "1.776315736"
    },
    {
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      "x": "0.621776579",
      "y": "1.762941251"
    },
    {
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      "x": "0.872459329",
      "y": "1.560208266"
    },
    {
      "turn": "-0.000088385",
      "x": "1.590572587",
      "y": "1.542338704"
    },
    {
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      "x": "1.340956749",
      "y": "1.542382828"
    },
    {
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      "x": "0.421100098",
      "y": "1.533875848"
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    {
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      "x": "1.216110617",
      "y": "1.326231234"
    },
    {
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      "x": "0.966494779",
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    {
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    {
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    {
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    {
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    {
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    {
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    {
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    {
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    {
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    {
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    {
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    {
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    {
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    }
  ],
  "radius": "0.249615841"
}

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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