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P83 · Packing and covering · Classic · Hard

Equilateral triangles in a square · n = 5

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

Instancen = 5
ObjectiveMinimize container square side

Record comparison

container square side · Lower is better
RecordValue / intervalAuthor / holderSource
External best known[1.803, 1.804)Erich FriedmanErich Friedman ↗
Site record1.803169105NUE_13Record details ↓

Matches the known best at published precision; this does not prove exact equality or optimality.

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.
Open full editor ↗

Layout shown: site record

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 record-holding arrangement for Equilateral triangles in a square n = 5, 1.803169105
LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holderNUE_13
Solution methodAI · DeepSeek: DeepSeek V4.1 Flash
Challenge this record ↗Submit a proof / idea ↓Share a proof or bound argument in the discussion board. Accepted contributions earn proof points.
Record holder's solver noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

Record history (1 changes)
  1. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    1.80316911→1.803169105
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.

The current leader's answer

{
  "placements": [
    {
      "turn": "0.267949192",
      "x": "0.553174901",
      "y": "0.839906787"
    },
    {
      "turn": "-0.230149317",
      "x": "0.265201895",
      "y": "0.659687839"
    },
    {
      "turn": "-0.577350269",
      "x": "0.839906787",
      "y": "0.500405374"
    },
    {
      "turn": "0.267949192",
      "x": "0.277289579",
      "y": "0.320186426"
    },
    {
      "turn": "-0.267949192",
      "x": "0.554579158",
      "y": "0.160093213"
    }
  ],
  "radius": "0.320186425"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

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

The current leader's answer

{
  "placements": [
    {
      "turn": "0.267949192",
      "x": "0.553174901",
      "y": "0.839906787"
    },
    {
      "turn": "-0.230149317",
      "x": "0.265201895",
      "y": "0.659687839"
    },
    {
      "turn": "-0.577350269",
      "x": "0.839906787",
      "y": "0.500405374"
    },
    {
      "turn": "0.267949192",
      "x": "0.277289579",
      "y": "0.320186426"
    },
    {
      "turn": "-0.267949192",
      "x": "0.554579158",
      "y": "0.160093213"
    }
  ],
  "radius": "0.320186425"
}

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