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P85 · Packing and covering · Classic · Hard

Squares in an equilateral triangle · n = 4

Pack n unit squares into an equilateral triangle container with boundary contact allowed and disjoint interiors, minimising the side length of the container triangle.

Instancen = 4
ObjectiveMinimize container triangle side

Record comparison

container triangle side · Lower is better
RecordValue / intervalAuthor / holderSource
External best known[4.154, 4.155)Erich FriedmanErich Friedman ↗
Site recordNo player record yet——

Formal definition

  • ContainerThe editor and certificate normalise the container to the equilateral triangle with vertices (0,0), (1,0), and (1/2,√3/2), 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 triangle side = 1/(2 sin(π/4) 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 Squares in an equilateral triangle n = 4
LAYOUT & HISTORY

Starter layout

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ANSWER FORMAT

How to write your answer

the equilateral triangle with vertices (0,0), (1,0), (1/2,√3/2)

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": "-0.414213562",
      "x": "0.259308766",
      "y": "0.120345617"
    },
    {
      "turn": "-0.414213562",
      "x": "0.500000000",
      "y": "0.120345617"
    },
    {
      "turn": "-0.414213562",
      "x": "0.740691234",
      "y": "0.120345617"
    },
    {
      "turn": "-0.414213562",
      "x": "0.500000000",
      "y": "0.361036851"
    }
  ],
  "radius": "0.170194403"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 4,
  "sides": 5
}

Starter answer

{
  "placements": [
    {
      "turn": "-0.414213562",
      "x": "0.259308766",
      "y": "0.120345617"
    },
    {
      "turn": "-0.414213562",
      "x": "0.500000000",
      "y": "0.120345617"
    },
    {
      "turn": "-0.414213562",
      "x": "0.740691234",
      "y": "0.120345617"
    },
    {
      "turn": "-0.414213562",
      "x": "0.500000000",
      "y": "0.361036851"
    }
  ],
  "radius": "0.170194403"
}

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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外部证明NUE_13#310 replies外部记录更新（9.23更新）NUE_13#310 replies