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P84 · Packing and covering · Classic · Hard

Equilateral triangles in a disc · n = 23

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

Instancen = 23
ObjectiveMinimize container circle radius
Best known, unproven1.98467Source-precision interval: 1.98467–1.98468 (values inside match)exhibited site certificate 1.984672742Compiled by Erich Friedman; Found by Jake Loyd in June 2026. Source expression: r = 1.98467+.

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 = 23, 1.984672742
Current leader

1.984672742

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": "0.302552234",
      "x": "0.771195531",
      "y": "1.742948006"
    },
    {
      "turn": "0.302552234",
      "x": "1.372174363",
      "y": "1.627816522"
    },
    {
      "turn": "-0.233934654",
      "x": "1.031912341",
      "y": "1.613904661"
    },
    {
      "turn": "-0.233934654",
      "x": "0.529082311",
      "y": "1.581682298"
    },
    {
      "turn": "-0.198839465",
      "x": "1.652298695",
      "y": "1.500671766"
    },
    {
      "turn": "0.302552234",
      "x": "1.050515931",
      "y": "1.323595608"
    },
    {
      "turn": "0.302552234",
      "x": "0.547685901",
      "y": "1.291373245"
    },
    {
      "turn": "-0.233934654",
      "x": "0.243993387",
      "y": "1.222558444"
    },
    {
      "turn": "-0.233934654",
      "x": "1.311232741",
      "y": "1.194552263"
    },
    {
      "turn": "-0.233934654",
      "x": "0.808402711",
      "y": "1.162329900"
    },
    {
      "turn": "-0.233934654",
      "x": "1.737680713",
      "y": "1.072453716"
    },
    {
      "turn": "0.302552234",
      "x": "0.262596977",
      "y": "0.932249391"
    },
    {
      "turn": "0.302552234",
      "x": "1.222946284",
      "y": "0.897393480"
    },
    {
      "turn": "0.302552234",
      "x": "0.720116254",
      "y": "0.865171117"
    },
    {
      "turn": "0.302552234",
      "x": "1.756284303",
      "y": "0.782144663"
    },
    {
      "turn": "-0.233934654",
      "x": "0.980833064",
      "y": "0.736127772"
    },
    {
      "turn": "-0.233934654",
      "x": "1.454794020",
      "y": "0.710023585"
    },
    {
      "turn": "-0.233934654",
      "x": "0.475800804",
      "y": "0.707211686"
    },
    {
      "turn": "0.302552234",
      "x": "0.999436654",
      "y": "0.445818719"
    },
    {
      "turn": "0.302552234",
      "x": "1.473397609",
      "y": "0.419714532"
    },
    {
      "turn": "0.302552234",
      "x": "0.494404394",
      "y": "0.416902632"
    },
    {
      "turn": "-0.233934654",
      "x": "0.755121204",
      "y": "0.287859287"
    },
    {
      "turn": "-0.233934654",
      "x": "1.231284389",
      "y": "0.258448824"
    }
  ],
  "radius": "0.290904519"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 23,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.302552234",
      "x": "0.771195531",
      "y": "1.742948006"
    },
    {
      "turn": "0.302552234",
      "x": "1.372174363",
      "y": "1.627816522"
    },
    {
      "turn": "-0.233934654",
      "x": "1.031912341",
      "y": "1.613904661"
    },
    {
      "turn": "-0.233934654",
      "x": "0.529082311",
      "y": "1.581682298"
    },
    {
      "turn": "-0.198839465",
      "x": "1.652298695",
      "y": "1.500671766"
    },
    {
      "turn": "0.302552234",
      "x": "1.050515931",
      "y": "1.323595608"
    },
    {
      "turn": "0.302552234",
      "x": "0.547685901",
      "y": "1.291373245"
    },
    {
      "turn": "-0.233934654",
      "x": "0.243993387",
      "y": "1.222558444"
    },
    {
      "turn": "-0.233934654",
      "x": "1.311232741",
      "y": "1.194552263"
    },
    {
      "turn": "-0.233934654",
      "x": "0.808402711",
      "y": "1.162329900"
    },
    {
      "turn": "-0.233934654",
      "x": "1.737680713",
      "y": "1.072453716"
    },
    {
      "turn": "0.302552234",
      "x": "0.262596977",
      "y": "0.932249391"
    },
    {
      "turn": "0.302552234",
      "x": "1.222946284",
      "y": "0.897393480"
    },
    {
      "turn": "0.302552234",
      "x": "0.720116254",
      "y": "0.865171117"
    },
    {
      "turn": "0.302552234",
      "x": "1.756284303",
      "y": "0.782144663"
    },
    {
      "turn": "-0.233934654",
      "x": "0.980833064",
      "y": "0.736127772"
    },
    {
      "turn": "-0.233934654",
      "x": "1.454794020",
      "y": "0.710023585"
    },
    {
      "turn": "-0.233934654",
      "x": "0.475800804",
      "y": "0.707211686"
    },
    {
      "turn": "0.302552234",
      "x": "0.999436654",
      "y": "0.445818719"
    },
    {
      "turn": "0.302552234",
      "x": "1.473397609",
      "y": "0.419714532"
    },
    {
      "turn": "0.302552234",
      "x": "0.494404394",
      "y": "0.416902632"
    },
    {
      "turn": "-0.233934654",
      "x": "0.755121204",
      "y": "0.287859287"
    },
    {
      "turn": "-0.233934654",
      "x": "1.231284389",
      "y": "0.258448824"
    }
  ],
  "radius": "0.290904519"
}

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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