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P84 · Packing and covering · Classic · Hard

Equilateral triangles in a disc · n = 18

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

Instancen = 18
ObjectiveMinimize container circle radius
Best known, unproven1.80176Source-precision interval: 1.80176–1.80177 (values inside match)exhibited site certificate 1.801764491Compiled by Erich Friedman; Found by Jake Loyd in June 2026. Source expression: r = 1.80176+.

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 = 18, 1.801764491
Current leader

1.801764491

container circle radius

Matches the best known
Answer sourceJake Loyd
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": "0.371235163",
      "x": "0.892481897",
      "y": "1.781708168"
    },
    {
      "turn": "-0.169735282",
      "x": "1.194971408",
      "y": "1.675975725"
    },
    {
      "turn": "0.371235163",
      "x": "0.503984897",
      "y": "1.496256245"
    },
    {
      "turn": "0.371235163",
      "x": "1.465869932",
      "y": "1.401183990"
    },
    {
      "turn": "-0.169735282",
      "x": "0.806474407",
      "y": "1.390523802"
    },
    {
      "turn": "-0.169735282",
      "x": "0.261173159",
      "y": "1.287158866"
    },
    {
      "turn": "-0.169735282",
      "x": "1.697423068",
      "y": "1.092509968"
    },
    {
      "turn": "0.371235163",
      "x": "1.101652326",
      "y": "1.087537922"
    },
    {
      "turn": "0.371235163",
      "x": "0.568628227",
      "y": "1.019296631"
    },
    {
      "turn": "-0.169735282",
      "x": "0.858840588",
      "y": "0.878440543"
    },
    {
      "turn": "-0.169735282",
      "x": "0.325816490",
      "y": "0.810199252"
    },
    {
      "turn": "0.371235163",
      "x": "1.757100842",
      "y": "0.777680145"
    },
    {
      "turn": "-0.169735282",
      "x": "1.357484857",
      "y": "0.750669753"
    },
    {
      "turn": "0.371235163",
      "x": "0.955074906",
      "y": "0.570540220"
    },
    {
      "turn": "0.371235163",
      "x": "0.409773658",
      "y": "0.467175284"
    },
    {
      "turn": "0.371235163",
      "x": "1.417162631",
      "y": "0.435839931"
    },
    {
      "turn": "-0.169735282",
      "x": "0.712263168",
      "y": "0.361442841"
    },
    {
      "turn": "-0.169735282",
      "x": "1.174350893",
      "y": "0.226742552"
    }
  ],
  "radius": "0.320436035"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 18,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.371235163",
      "x": "0.892481897",
      "y": "1.781708168"
    },
    {
      "turn": "-0.169735282",
      "x": "1.194971408",
      "y": "1.675975725"
    },
    {
      "turn": "0.371235163",
      "x": "0.503984897",
      "y": "1.496256245"
    },
    {
      "turn": "0.371235163",
      "x": "1.465869932",
      "y": "1.401183990"
    },
    {
      "turn": "-0.169735282",
      "x": "0.806474407",
      "y": "1.390523802"
    },
    {
      "turn": "-0.169735282",
      "x": "0.261173159",
      "y": "1.287158866"
    },
    {
      "turn": "-0.169735282",
      "x": "1.697423068",
      "y": "1.092509968"
    },
    {
      "turn": "0.371235163",
      "x": "1.101652326",
      "y": "1.087537922"
    },
    {
      "turn": "0.371235163",
      "x": "0.568628227",
      "y": "1.019296631"
    },
    {
      "turn": "-0.169735282",
      "x": "0.858840588",
      "y": "0.878440543"
    },
    {
      "turn": "-0.169735282",
      "x": "0.325816490",
      "y": "0.810199252"
    },
    {
      "turn": "0.371235163",
      "x": "1.757100842",
      "y": "0.777680145"
    },
    {
      "turn": "-0.169735282",
      "x": "1.357484857",
      "y": "0.750669753"
    },
    {
      "turn": "0.371235163",
      "x": "0.955074906",
      "y": "0.570540220"
    },
    {
      "turn": "0.371235163",
      "x": "0.409773658",
      "y": "0.467175284"
    },
    {
      "turn": "0.371235163",
      "x": "1.417162631",
      "y": "0.435839931"
    },
    {
      "turn": "-0.169735282",
      "x": "0.712263168",
      "y": "0.361442841"
    },
    {
      "turn": "-0.169735282",
      "x": "1.174350893",
      "y": "0.226742552"
    }
  ],
  "radius": "0.320436035"
}

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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