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P84 · Packing and covering · Classic · Hard

Equilateral triangles in a disc · n = 22

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

Instancen = 22
ObjectiveMinimize container circle radius

Record comparison

container circle radius · Lower is better
RecordValue / intervalAuthor / holderSource
External best known[1.943, 1.944)Maurizio MorandiErich 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 = 22
LAYOUT & HISTORY

Starter layout

Inspect the layout, solver notes and record history.

Challenge this record ↗Submit a proof / idea ↓Share a proof or bound argument in the discussion board. Accepted contributions earn proof points.
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": "0.268200414",
      "x": "0.867571454",
      "y": "1.772565558"
    },
    {
      "turn": "0.268200414",
      "x": "1.321102191",
      "y": "1.667252860"
    },
    {
      "turn": "-0.267698002",
      "x": "0.610437905",
      "y": "1.623948680"
    },
    {
      "turn": "-0.267698002",
      "x": "1.063968642",
      "y": "1.518635982"
    },
    {
      "turn": "-0.267698002",
      "x": "1.578374957",
      "y": "1.518877113"
    },
    {
      "turn": "0.268200414",
      "x": "0.514707568",
      "y": "1.326911116"
    },
    {
      "turn": "0.268200414",
      "x": "1.482644620",
      "y": "1.221839549"
    },
    {
      "turn": "0.268200414",
      "x": "0.968238305",
      "y": "1.221598418"
    },
    {
      "turn": "-0.267698002",
      "x": "0.257574019",
      "y": "1.178294239"
    },
    {
      "turn": "-0.267698002",
      "x": "1.739917387",
      "y": "1.073463802"
    },
    {
      "turn": "-0.267698002",
      "x": "1.225511071",
      "y": "1.073222671"
    },
    {
      "turn": "-0.267698002",
      "x": "0.711104756",
      "y": "1.072981540"
    },
    {
      "turn": "0.268200414",
      "x": "0.257713236",
      "y": "0.881301614"
    },
    {
      "turn": "0.268200414",
      "x": "1.740056604",
      "y": "0.776471178"
    },
    {
      "turn": "0.268200414",
      "x": "0.711243973",
      "y": "0.775988916"
    },
    {
      "turn": "0.268200414",
      "x": "1.225650289",
      "y": "0.776230047"
    },
    {
      "turn": "-0.267698002",
      "x": "1.482923055",
      "y": "0.627854300"
    },
    {
      "turn": "-0.267698002",
      "x": "0.968516740",
      "y": "0.627613169"
    },
    {
      "turn": "-0.267698002",
      "x": "0.454110424",
      "y": "0.627372038"
    },
    {
      "turn": "0.268200414",
      "x": "0.743110611",
      "y": "0.330514819"
    },
    {
      "turn": "0.268200414",
      "x": "1.257516926",
      "y": "0.330755950"
    },
    {
      "turn": "-0.267698002",
      "x": "1.000383377",
      "y": "0.182139072"
    }
  ],
  "radius": "0.296992656"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 22,
  "sides": 5
}

Starter answer

{
  "placements": [
    {
      "turn": "0.268200414",
      "x": "0.867571454",
      "y": "1.772565558"
    },
    {
      "turn": "0.268200414",
      "x": "1.321102191",
      "y": "1.667252860"
    },
    {
      "turn": "-0.267698002",
      "x": "0.610437905",
      "y": "1.623948680"
    },
    {
      "turn": "-0.267698002",
      "x": "1.063968642",
      "y": "1.518635982"
    },
    {
      "turn": "-0.267698002",
      "x": "1.578374957",
      "y": "1.518877113"
    },
    {
      "turn": "0.268200414",
      "x": "0.514707568",
      "y": "1.326911116"
    },
    {
      "turn": "0.268200414",
      "x": "1.482644620",
      "y": "1.221839549"
    },
    {
      "turn": "0.268200414",
      "x": "0.968238305",
      "y": "1.221598418"
    },
    {
      "turn": "-0.267698002",
      "x": "0.257574019",
      "y": "1.178294239"
    },
    {
      "turn": "-0.267698002",
      "x": "1.739917387",
      "y": "1.073463802"
    },
    {
      "turn": "-0.267698002",
      "x": "1.225511071",
      "y": "1.073222671"
    },
    {
      "turn": "-0.267698002",
      "x": "0.711104756",
      "y": "1.072981540"
    },
    {
      "turn": "0.268200414",
      "x": "0.257713236",
      "y": "0.881301614"
    },
    {
      "turn": "0.268200414",
      "x": "1.740056604",
      "y": "0.776471178"
    },
    {
      "turn": "0.268200414",
      "x": "0.711243973",
      "y": "0.775988916"
    },
    {
      "turn": "0.268200414",
      "x": "1.225650289",
      "y": "0.776230047"
    },
    {
      "turn": "-0.267698002",
      "x": "1.482923055",
      "y": "0.627854300"
    },
    {
      "turn": "-0.267698002",
      "x": "0.968516740",
      "y": "0.627613169"
    },
    {
      "turn": "-0.267698002",
      "x": "0.454110424",
      "y": "0.627372038"
    },
    {
      "turn": "0.268200414",
      "x": "0.743110611",
      "y": "0.330514819"
    },
    {
      "turn": "0.268200414",
      "x": "1.257516926",
      "y": "0.330755950"
    },
    {
      "turn": "-0.267698002",
      "x": "1.000383377",
      "y": "0.182139072"
    }
  ],
  "radius": "0.296992656"
}

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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