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P84 · Packing and covering · Classic · Hard

Equilateral triangles in a disc · n = 14

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

Instancen = 14
ObjectiveMinimize container circle radius

Record comparison

container circle radius · Lower is better
RecordValue / intervalAuthor / holderSource
External best known[1.60448, 1.60449)Jake LoydErich 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.
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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 = 14
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": "0.047245168",
      "x": "1.320535342",
      "y": "1.700428816"
    },
    {
      "turn": "0.568528850",
      "x": "0.927999297",
      "y": "1.684495387"
    },
    {
      "turn": "0.568528850",
      "x": "0.426672860",
      "y": "1.354760795"
    },
    {
      "turn": "-0.006641433",
      "x": "0.786475311",
      "y": "1.349981376"
    },
    {
      "turn": "-0.045936025",
      "x": "1.616453928",
      "y": "1.256188822"
    },
    {
      "turn": "0.567291188",
      "x": "1.232183174",
      "y": "1.186434091"
    },
    {
      "turn": "-0.025441396",
      "x": "0.230708626",
      "y": "1.039169046"
    },
    {
      "turn": "0.517684638",
      "x": "1.767043937",
      "y": "0.929381087"
    },
    {
      "turn": "-0.490378552",
      "x": "0.736170486",
      "y": "0.858833169"
    },
    {
      "turn": "0.067781427",
      "x": "1.107274952",
      "y": "0.800468326"
    },
    {
      "turn": "0.067781427",
      "x": "0.488074604",
      "y": "0.571536995"
    },
    {
      "turn": "0.162191911",
      "x": "1.576246247",
      "y": "0.488840832"
    },
    {
      "turn": "-0.379610998",
      "x": "1.234858549",
      "y": "0.375108314"
    },
    {
      "turn": "-0.490378552",
      "x": "0.708397718",
      "y": "0.287040295"
    }
  ],
  "radius": "0.359834192"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 14,
  "sides": 5
}

Starter answer

{
  "placements": [
    {
      "turn": "0.047245168",
      "x": "1.320535342",
      "y": "1.700428816"
    },
    {
      "turn": "0.568528850",
      "x": "0.927999297",
      "y": "1.684495387"
    },
    {
      "turn": "0.568528850",
      "x": "0.426672860",
      "y": "1.354760795"
    },
    {
      "turn": "-0.006641433",
      "x": "0.786475311",
      "y": "1.349981376"
    },
    {
      "turn": "-0.045936025",
      "x": "1.616453928",
      "y": "1.256188822"
    },
    {
      "turn": "0.567291188",
      "x": "1.232183174",
      "y": "1.186434091"
    },
    {
      "turn": "-0.025441396",
      "x": "0.230708626",
      "y": "1.039169046"
    },
    {
      "turn": "0.517684638",
      "x": "1.767043937",
      "y": "0.929381087"
    },
    {
      "turn": "-0.490378552",
      "x": "0.736170486",
      "y": "0.858833169"
    },
    {
      "turn": "0.067781427",
      "x": "1.107274952",
      "y": "0.800468326"
    },
    {
      "turn": "0.067781427",
      "x": "0.488074604",
      "y": "0.571536995"
    },
    {
      "turn": "0.162191911",
      "x": "1.576246247",
      "y": "0.488840832"
    },
    {
      "turn": "-0.379610998",
      "x": "1.234858549",
      "y": "0.375108314"
    },
    {
      "turn": "-0.490378552",
      "x": "0.708397718",
      "y": "0.287040295"
    }
  ],
  "radius": "0.359834192"
}

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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