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P84 · Packing and covering · Classic · Hard

Equilateral triangles in a disc · n = 30

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

Instancen = 30
ObjectiveMinimize container circle radius
Best known, unproven2.25124Source-precision interval: 2.25124–2.25125 (values inside match)exhibited site certificate 2.251247456Compiled by Erich Friedman; Found by Thomas Schadt in August 2026. Source expression: r = 2.25124+.

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 = 30, 2.251247456
Current leader

2.251247456

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

{
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      "turn": "0.169603979",
      "x": "0.887684917",
      "y": "1.784438329"
    },
    {
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      "y": "1.751379926"
    },
    {
      "turn": "-0.371380387",
      "x": "0.645568721",
      "y": "1.699878173"
    },
    {
      "turn": "-0.371380387",
      "x": "1.081974259",
      "y": "1.617039631"
    },
    {
      "turn": "-0.371380387",
      "x": "1.500993862",
      "y": "1.583981228"
    },
    {
      "turn": "-0.371380387",
      "x": "0.318837817",
      "y": "1.468691691"
    },
    {
      "turn": "0.169603979",
      "x": "0.597741867",
      "y": "1.447919318"
    },
    {
      "turn": "0.169603979",
      "x": "1.034147404",
      "y": "1.365080777"
    },
    {
      "turn": "0.169603979",
      "x": "1.504970839",
      "y": "1.322188965"
    },
    {
      "turn": "-0.371380387",
      "x": "0.792031209",
      "y": "1.280520620"
    },
    {
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      "x": "1.800227353",
      "y": "1.271976635"
    },
    {
      "turn": "-0.371380387",
      "x": "1.262854643",
      "y": "1.237628809"
    },
    {
      "turn": "0.169603979",
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      "y": "1.216732837"
    },
    {
      "turn": "-0.371380387",
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    },
    {
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      "x": "0.744204354",
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    {
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      "x": "1.752400498",
      "y": "1.020017781"
    },
    {
      "turn": "0.169603979",
      "x": "1.215027789",
      "y": "0.985669954"
    },
    {
      "turn": "-0.371380387",
      "x": "1.510284303",
      "y": "0.935457624"
    },
    {
      "turn": "-0.371380387",
      "x": "0.972911593",
      "y": "0.901109798"
    },
    {
      "turn": "0.077458159",
      "x": "0.172694927",
      "y": "0.846815604"
    },
    {
      "turn": "0.030541448",
      "x": "0.608933141",
      "y": "0.676918816"
    },
    {
      "turn": "-0.478493641",
      "x": "0.333595955",
      "y": "0.647111995"
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    {
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      "y": "0.609844483"
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    {
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      "y": "0.553969138"
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    {
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      "x": "0.890040716",
      "y": "0.525284327"
    },
    {
      "turn": "0.513234510",
      "x": "0.506303354",
      "y": "0.312013359"
    },
    {
      "turn": "0.137501628",
      "x": "1.396425506",
      "y": "0.305503493"
    },
    {
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      "x": "0.842213861",
      "y": "0.273325472"
    },
    {
      "turn": "-0.371380387",
      "x": "1.138610613",
      "y": "0.224436546"
    }
  ],
  "radius": "0.256457933"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 30,
  "sides": 5
}

The current leader's answer

{
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      "x": "0.887684917",
      "y": "1.784438329"
    },
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    },
    {
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      "x": "0.645568721",
      "y": "1.699878173"
    },
    {
      "turn": "-0.371380387",
      "x": "1.081974259",
      "y": "1.617039631"
    },
    {
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    },
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      "y": "1.468691691"
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    {
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    {
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    }
  ],
  "radius": "0.256457933"
}

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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