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P86 · Packing and covering · Classic · Hard

Regular pentagons in a square · n = 11

Pack n unit-side regular pentagons 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[5.11555, 5.11556)Jonathan ViqueratErich Friedman ↗
Site record5.115560676zzzcyRecord details ↓

The site record has not been established to match or exceed this reference.

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/5) 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 Regular pentagons in a square n = 11, 5.115560676
LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holderzzzcy
Solution methodAI · OpenAI: GPT-6 Astra
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 (10 changes)
  1. zzzcyAI · OpenAI: GPT-6 Astra
    5.119837139→5.115560676
  2. NUE_13Human
    5.119838803→5.119837139
  3. NUE_13Human
    5.119855566→5.119838803
  4. NUE_13Human
    5.127792704→5.119855566
  5. NUE_13Human
    5.169951697→5.127792704
  6. hadsajncHuman
    5.199823742→5.169951697
  7. NUE_13Human
    5.21546997→5.199823742
  8. NUE_13Human
    5.250930884→5.21546997
  9. XSsMCHuman
    5.67100539→5.250930884
  10. XSsMCHuman
    9.357158883→5.67100539
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": "0.000000000",
      "x": "0.621115915",
      "y": "0.427206140"
    },
    {
      "turn": "-0.071929705",
      "x": "0.168594110",
      "y": "0.831713373"
    },
    {
      "turn": "0.026323705",
      "x": "0.141058627",
      "y": "0.159638601"
    },
    {
      "turn": "-0.158384440",
      "x": "0.541035885",
      "y": "0.865471058"
    },
    {
      "turn": "-0.324919696",
      "x": "0.352058031",
      "y": "0.334495112"
    },
    {
      "turn": "0.158384440",
      "x": "0.574760400",
      "y": "0.134528942"
    },
    {
      "turn": "0.000000000",
      "x": "0.134528942",
      "y": "0.492840620"
    },
    {
      "turn": "0.324919696",
      "x": "0.865471058",
      "y": "0.232332239"
    },
    {
      "turn": "-0.324919696",
      "x": "0.865471058",
      "y": "0.548628751"
    },
    {
      "turn": "0.158384440",
      "x": "0.384638440",
      "y": "0.646526619"
    },
    {
      "turn": "0.000000000",
      "x": "0.833713083",
      "y": "0.841851744"
    }
  ],
  "radius": "0.166286916"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

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

The current leader's answer

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "0.621115915",
      "y": "0.427206140"
    },
    {
      "turn": "-0.071929705",
      "x": "0.168594110",
      "y": "0.831713373"
    },
    {
      "turn": "0.026323705",
      "x": "0.141058627",
      "y": "0.159638601"
    },
    {
      "turn": "-0.158384440",
      "x": "0.541035885",
      "y": "0.865471058"
    },
    {
      "turn": "-0.324919696",
      "x": "0.352058031",
      "y": "0.334495112"
    },
    {
      "turn": "0.158384440",
      "x": "0.574760400",
      "y": "0.134528942"
    },
    {
      "turn": "0.000000000",
      "x": "0.134528942",
      "y": "0.492840620"
    },
    {
      "turn": "0.324919696",
      "x": "0.865471058",
      "y": "0.232332239"
    },
    {
      "turn": "-0.324919696",
      "x": "0.865471058",
      "y": "0.548628751"
    },
    {
      "turn": "0.158384440",
      "x": "0.384638440",
      "y": "0.646526619"
    },
    {
      "turn": "0.000000000",
      "x": "0.833713083",
      "y": "0.841851744"
    }
  ],
  "radius": "0.166286916"
}

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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外部记录更新NUE_13#311 replies