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P84 · Packing and covering · Classic · Hard

Equilateral triangles in a disc · n = 26

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

Instancen = 26
ObjectiveMinimize container circle radius
Best known, unproven2.10664Source-precision interval: 2.10664–2.10665 (values inside match)exhibited site certificate 2.106647674Compiled by Erich Friedman; Found by Thomas Schadt in August 2026. Source expression: r = 2.10664+.

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 = 26, 2.106647674
Current leader

2.106647674

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": "0.035143881",
      "x": "1.014152943",
      "y": "1.763125436"
    },
    {
      "turn": "-0.531423627",
      "x": "0.740767945",
      "y": "1.743886054"
    },
    {
      "turn": "0.035143881",
      "x": "1.431394396",
      "y": "1.690187150"
    },
    {
      "turn": "-0.531423627",
      "x": "1.167507236",
      "y": "1.535986774"
    },
    {
      "turn": "0.035143881",
      "x": "0.650907289",
      "y": "1.482809734"
    },
    {
      "turn": "-0.531423627",
      "x": "0.377522291",
      "y": "1.463570352"
    },
    {
      "turn": "-0.531423627",
      "x": "1.584748689",
      "y": "1.463048488"
    },
    {
      "turn": "0.035143881",
      "x": "1.077646580",
      "y": "1.274910454"
    },
    {
      "turn": "-0.531423627",
      "x": "0.804261582",
      "y": "1.255671072"
    },
    {
      "turn": "0.035143881",
      "x": "0.257491586",
      "y": "1.217192308"
    },
    {
      "turn": "0.035143881",
      "x": "1.494888034",
      "y": "1.201972169"
    },
    {
      "turn": "-0.503201998",
      "x": "1.828904255",
      "y": "1.095566445"
    },
    {
      "turn": "-0.531423627",
      "x": "1.231000873",
      "y": "1.047771792"
    },
    {
      "turn": "0.035143881",
      "x": "0.684230877",
      "y": "1.009293028"
    },
    {
      "turn": "-0.531423627",
      "x": "0.410845879",
      "y": "0.990053645"
    },
    {
      "turn": "-0.191563755",
      "x": "1.588969649",
      "y": "0.802224198"
    },
    {
      "turn": "0.035143881",
      "x": "1.110970167",
      "y": "0.801393748"
    },
    {
      "turn": "-0.531423627",
      "x": "0.837585169",
      "y": "0.782154366"
    },
    {
      "turn": "0.035143881",
      "x": "0.242234947",
      "y": "0.767342965"
    },
    {
      "turn": "0.035143881",
      "x": "0.668974238",
      "y": "0.559443685"
    },
    {
      "turn": "-0.531423627",
      "x": "0.395589240",
      "y": "0.540204303"
    },
    {
      "turn": "0.347367844",
      "x": "1.633649256",
      "y": "0.531781157"
    },
    {
      "turn": "0.547943763",
      "x": "1.134178405",
      "y": "0.474989860"
    },
    {
      "turn": "-0.022339335",
      "x": "1.399476788",
      "y": "0.272836882"
    },
    {
      "turn": "-0.022339335",
      "x": "0.986685613",
      "y": "0.244002009"
    },
    {
      "turn": "-0.531423627",
      "x": "0.685838527",
      "y": "0.240152821"
    }
  ],
  "radius": "0.274061143"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 26,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.035143881",
      "x": "1.014152943",
      "y": "1.763125436"
    },
    {
      "turn": "-0.531423627",
      "x": "0.740767945",
      "y": "1.743886054"
    },
    {
      "turn": "0.035143881",
      "x": "1.431394396",
      "y": "1.690187150"
    },
    {
      "turn": "-0.531423627",
      "x": "1.167507236",
      "y": "1.535986774"
    },
    {
      "turn": "0.035143881",
      "x": "0.650907289",
      "y": "1.482809734"
    },
    {
      "turn": "-0.531423627",
      "x": "0.377522291",
      "y": "1.463570352"
    },
    {
      "turn": "-0.531423627",
      "x": "1.584748689",
      "y": "1.463048488"
    },
    {
      "turn": "0.035143881",
      "x": "1.077646580",
      "y": "1.274910454"
    },
    {
      "turn": "-0.531423627",
      "x": "0.804261582",
      "y": "1.255671072"
    },
    {
      "turn": "0.035143881",
      "x": "0.257491586",
      "y": "1.217192308"
    },
    {
      "turn": "0.035143881",
      "x": "1.494888034",
      "y": "1.201972169"
    },
    {
      "turn": "-0.503201998",
      "x": "1.828904255",
      "y": "1.095566445"
    },
    {
      "turn": "-0.531423627",
      "x": "1.231000873",
      "y": "1.047771792"
    },
    {
      "turn": "0.035143881",
      "x": "0.684230877",
      "y": "1.009293028"
    },
    {
      "turn": "-0.531423627",
      "x": "0.410845879",
      "y": "0.990053645"
    },
    {
      "turn": "-0.191563755",
      "x": "1.588969649",
      "y": "0.802224198"
    },
    {
      "turn": "0.035143881",
      "x": "1.110970167",
      "y": "0.801393748"
    },
    {
      "turn": "-0.531423627",
      "x": "0.837585169",
      "y": "0.782154366"
    },
    {
      "turn": "0.035143881",
      "x": "0.242234947",
      "y": "0.767342965"
    },
    {
      "turn": "0.035143881",
      "x": "0.668974238",
      "y": "0.559443685"
    },
    {
      "turn": "-0.531423627",
      "x": "0.395589240",
      "y": "0.540204303"
    },
    {
      "turn": "0.347367844",
      "x": "1.633649256",
      "y": "0.531781157"
    },
    {
      "turn": "0.547943763",
      "x": "1.134178405",
      "y": "0.474989860"
    },
    {
      "turn": "-0.022339335",
      "x": "1.399476788",
      "y": "0.272836882"
    },
    {
      "turn": "-0.022339335",
      "x": "0.986685613",
      "y": "0.244002009"
    },
    {
      "turn": "-0.531423627",
      "x": "0.685838527",
      "y": "0.240152821"
    }
  ],
  "radius": "0.274061143"
}

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

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