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P85 · Packing and covering · Classic · Hard

Squares in an equilateral triangle · n = 9

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 = 9
ObjectiveMinimize container triangle side

Record comparison

container triangle side · Lower is better
RecordValue / intervalAuthor / holderSource
External best known[5.309, 5.31)Erich FriedmanErich Friedman ↗
Site record5.309401096condorRecord 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 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.
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 Squares in an equilateral triangle n = 9, 5.309401096
LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holdercondor
Solution methodAI · OpenAI: GPT-6.1 Sol
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. condorAI · OpenAI: GPT-6.1 Sol
    5.309401136→5.309401096
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.

The current leader's answer

{
  "placements": [
    {
      "turn": "-1.303225371",
      "x": "0.519901008",
      "y": "0.643210686"
    },
    {
      "turn": "1.303225371",
      "x": "0.371357864",
      "y": "0.454865524"
    },
    {
      "turn": "-1.303225375",
      "x": "0.614073588",
      "y": "0.480098992"
    },
    {
      "turn": "1.303225372",
      "x": "0.183012702",
      "y": "0.128642137"
    },
    {
      "turn": "-2.414213564",
      "x": "0.420395968",
      "y": "0.094172581"
    },
    {
      "turn": "1.303225375",
      "x": "0.277185282",
      "y": "0.291753831"
    },
    {
      "turn": "-2.414213561",
      "x": "0.797086290",
      "y": "0.094172581"
    },
    {
      "turn": "-1.303225375",
      "x": "0.708246168",
      "y": "0.316987298"
    },
    {
      "turn": "2.414213560",
      "x": "0.608741129",
      "y": "0.094172581"
    }
  ],
  "radius": "0.133180140"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 9,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "-1.303225371",
      "x": "0.519901008",
      "y": "0.643210686"
    },
    {
      "turn": "1.303225371",
      "x": "0.371357864",
      "y": "0.454865524"
    },
    {
      "turn": "-1.303225375",
      "x": "0.614073588",
      "y": "0.480098992"
    },
    {
      "turn": "1.303225372",
      "x": "0.183012702",
      "y": "0.128642137"
    },
    {
      "turn": "-2.414213564",
      "x": "0.420395968",
      "y": "0.094172581"
    },
    {
      "turn": "1.303225375",
      "x": "0.277185282",
      "y": "0.291753831"
    },
    {
      "turn": "-2.414213561",
      "x": "0.797086290",
      "y": "0.094172581"
    },
    {
      "turn": "-1.303225375",
      "x": "0.708246168",
      "y": "0.316987298"
    },
    {
      "turn": "2.414213560",
      "x": "0.608741129",
      "y": "0.094172581"
    }
  ],
  "radius": "0.133180140"
}

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