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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

Record comparison

container circle radius · Lower is better
RecordValue / intervalAuthor / holderSource
External best known[1.80176, 1.80177)Jake LoydErich Friedman ↗
Site record1.801764485NUE_13Record details ↓

Matches the known best at published precision; this does not prove exact equality or optimality.

Formal definition

  • ContainerThe editor and certificate normalise the container to the closed unit disc centred at (1,1), while public scores are reported in the literature's unit-piece scale. Boundary contact is allowed.
  • SubmissionSubmit {radius, placements:[{x,y,turn},…]}, with each value written as a decimal string of at most nine places. Here x, y are centre coordinates; circles require turn = 0 and polygons use turn = tan(θ/2). Coordinates and turn must lie in [−4, 4], with 0 < radius ≤ 4.
  • ConstraintsExactly n congruent pieces lie wholly inside the container with pairwise disjoint interiors. Polygons may rotate independently by arbitrary angles.
  • PrecisionInput decimals are parsed as exact rationals and regular-polygon vertices are defined algebraically, with zero floating-point tolerance in feasibility checks.
  • ObjectiveContainer circle radius = 1/(2 sin(π/3) r), where smaller is better. Pages, leaderboards, and literature records share this unit with automatic conversion.
Open full editor ↗

Layout shown: site record

VERIFIED CONSTRUCTIONVERIFIED CONSTRUCTION

Getting a feel for it

Literature targets and coordinate certificates

Literature targets are converted at their published precision, while nine-decimal coordinate certificates are stored and displayed alongside the source values.

Source
The record-holding arrangement for Equilateral triangles in a disc n = 18, 1.801764485
LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holderNUE_13
Solution methodAI · DeepSeek: DeepSeek V4.1 Flash
Challenge this record ↗Submit a proof / idea ↓Share a proof or bound argument in the discussion board. Accepted contributions earn proof points.
Record holder's solver noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

Record history (1 changes)
  1. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    1.801764491→1.801764485
ANSWER FORMAT

How to write your answer

the closed unit disc centred at (1,1)

Submit {radius, placements:[{x,y,turn},…]}, with each value written as a decimal string of at most nine places. Here x, y are centre coordinates; circles require turn = 0 and polygons use turn = tan(θ/2). Coordinates and turn must lie in [−4, 4], with 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": "0.503984896",
      "y": "1.496256245"
    },
    {
      "turn": "0.371235163",
      "x": "1.465869931",
      "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": "0.858840589",
      "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": "0.955074905",
      "y": "0.570540220"
    },
    {
      "turn": "0.371235163",
      "x": "0.409773657",
      "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": "0.320436036"
}
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": "0.503984896",
      "y": "1.496256245"
    },
    {
      "turn": "0.371235163",
      "x": "1.465869931",
      "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": "0.858840589",
      "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": "0.955074905",
      "y": "0.570540220"
    },
    {
      "turn": "0.371235163",
      "x": "0.409773657",
      "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": "0.320436036"
}

Submit {radius, placements:[{x,y,turn},…]}, with each value written as a decimal string of at most nine places. Here x, y are centre coordinates; circles require turn = 0 and polygons use turn = tan(θ/2). Coordinates and turn must lie in [−4, 4], with 0 < radius ≤ 4. · Verifier v1.0.0

DISCUSSION

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