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P83 · Packing and covering · Classic · Hard

Equilateral triangles in a square · n = 11

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

Instancen = 11
ObjectiveMinimize container square side

Record comparison

container square side · Lower is better
RecordValue / intervalAuthor / holderSource
External best known[2.49, 2.491)Maurizio MorandiErich Friedman ↗
Site recordNo player record yet——

Formal definition

  • ContainerThe editor and certificate normalise the container to the unit square [0,1]², 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 square side = 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 square n = 11
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 unit square [0,1]²

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": "0.267949192",
      "x": "0.515787640",
      "y": "0.884110848"
    },
    {
      "turn": "0.490962183",
      "x": "0.858256770",
      "y": "0.816615052"
    },
    {
      "turn": "-0.484525357",
      "x": "0.229352203",
      "y": "0.784651311"
    },
    {
      "turn": "-0.067308898",
      "x": "0.716513540",
      "y": "0.633230104"
    },
    {
      "turn": "0.072534148",
      "x": "0.143642667",
      "y": "0.569302622"
    },
    {
      "turn": "-0.285417374",
      "x": "0.427425173",
      "y": "0.470011513"
    },
    {
      "turn": "-0.577350269",
      "x": "0.884110848",
      "y": "0.432504204"
    },
    {
      "turn": "0.267949192",
      "x": "0.200725900",
      "y": "0.238064781"
    },
    {
      "turn": "0.267949192",
      "x": "0.598548200",
      "y": "0.231778305"
    },
    {
      "turn": "-0.267949192",
      "x": "0.397822301",
      "y": "0.115889152"
    },
    {
      "turn": "-0.267949192",
      "x": "0.799274100",
      "y": "0.115889152"
    }
  ],
  "radius": "0.231778303"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 11,
  "sides": 5
}

Starter answer

{
  "placements": [
    {
      "turn": "0.267949192",
      "x": "0.515787640",
      "y": "0.884110848"
    },
    {
      "turn": "0.490962183",
      "x": "0.858256770",
      "y": "0.816615052"
    },
    {
      "turn": "-0.484525357",
      "x": "0.229352203",
      "y": "0.784651311"
    },
    {
      "turn": "-0.067308898",
      "x": "0.716513540",
      "y": "0.633230104"
    },
    {
      "turn": "0.072534148",
      "x": "0.143642667",
      "y": "0.569302622"
    },
    {
      "turn": "-0.285417374",
      "x": "0.427425173",
      "y": "0.470011513"
    },
    {
      "turn": "-0.577350269",
      "x": "0.884110848",
      "y": "0.432504204"
    },
    {
      "turn": "0.267949192",
      "x": "0.200725900",
      "y": "0.238064781"
    },
    {
      "turn": "0.267949192",
      "x": "0.598548200",
      "y": "0.231778305"
    },
    {
      "turn": "-0.267949192",
      "x": "0.397822301",
      "y": "0.115889152"
    },
    {
      "turn": "-0.267949192",
      "x": "0.799274100",
      "y": "0.115889152"
    }
  ],
  "radius": "0.231778303"
}

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

Discussion

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证书归一化半径居然是中心到顶点的距离NUE_13#312 replies