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P84 · Packing and covering · Classic · Hard

Equilateral triangles in a disc · n = 9

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

Instancen = 9
ObjectiveMinimize container circle radius

Record comparison

container circle radius · Lower is better
RecordValue / intervalAuthor / holderSource
External best known[1.314, 1.315)David W. CantrellErich Friedman ↗
Site recordNo player record yet——

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: starter layout

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 current arrangement for Equilateral triangles in a disc n = 9
LAYOUT & HISTORY

Starter layout

Inspect the layout, solver notes and record history.

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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.

Starter answer

{
  "placements": [
    {
      "turn": "0.067216310",
      "x": "1.267733489",
      "y": "1.652504971"
    },
    {
      "turn": "0.492010729",
      "x": "0.731284965",
      "y": "1.652101364"
    },
    {
      "turn": "-0.281632044",
      "x": "1.625376026",
      "y": "1.221005189"
    },
    {
      "turn": "-0.263855583",
      "x": "0.378687446",
      "y": "1.210004263"
    },
    {
      "turn": "0.268352427",
      "x": "0.995071933",
      "y": "1.026596044"
    },
    {
      "turn": "0.204055598",
      "x": "0.351090320",
      "y": "0.723666532"
    },
    {
      "turn": "0.336341958",
      "x": "1.650091477",
      "y": "0.726458358"
    },
    {
      "turn": "-0.490210027",
      "x": "0.733178215",
      "y": "0.347121685"
    },
    {
      "turn": "-0.065759108",
      "x": "1.269625652",
      "y": "0.348274623"
    }
  ],
  "radius": "0.439132387"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 9,
  "sides": 5
}

Starter answer

{
  "placements": [
    {
      "turn": "0.067216310",
      "x": "1.267733489",
      "y": "1.652504971"
    },
    {
      "turn": "0.492010729",
      "x": "0.731284965",
      "y": "1.652101364"
    },
    {
      "turn": "-0.281632044",
      "x": "1.625376026",
      "y": "1.221005189"
    },
    {
      "turn": "-0.263855583",
      "x": "0.378687446",
      "y": "1.210004263"
    },
    {
      "turn": "0.268352427",
      "x": "0.995071933",
      "y": "1.026596044"
    },
    {
      "turn": "0.204055598",
      "x": "0.351090320",
      "y": "0.723666532"
    },
    {
      "turn": "0.336341958",
      "x": "1.650091477",
      "y": "0.726458358"
    },
    {
      "turn": "-0.490210027",
      "x": "0.733178215",
      "y": "0.347121685"
    },
    {
      "turn": "-0.065759108",
      "x": "1.269625652",
      "y": "0.348274623"
    }
  ],
  "radius": "0.439132387"
}

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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