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P86 · Packing and covering · Classic · Hard

Regular pentagons in a square · n = 16

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

Instancen = 16
ObjectiveMinimize container square side

Record comparison

container square side · Lower is better
RecordValue / intervalAuthor / holderSource
External best known[6.06451, 6.06452)Aapo Lipponen2026-10Erich 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/5) 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 Regular pentagons in a square n = 16
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": "-0.076289847",
      "x": "0.635518039",
      "y": "0.874717465"
    },
    {
      "turn": "0.242615877",
      "x": "0.138643502",
      "y": "0.861566923"
    },
    {
      "turn": "-0.076289840",
      "x": "0.370275505",
      "y": "0.874717464"
    },
    {
      "turn": "-0.324919696",
      "x": "0.886521776",
      "y": "0.815029350"
    },
    {
      "turn": "0.242615877",
      "x": "0.503748883",
      "y": "0.686961985"
    },
    {
      "turn": "-0.030639622",
      "x": "0.732201065",
      "y": "0.633086977"
    },
    {
      "turn": "-0.076289840",
      "x": "0.124671857",
      "y": "0.629322530"
    },
    {
      "turn": "0.242615877",
      "x": "0.310847496",
      "y": "0.494777157"
    },
    {
      "turn": "-0.076289840",
      "x": "0.535458071",
      "y": "0.462177428"
    },
    {
      "turn": "0.324919696",
      "x": "0.886521776",
      "y": "0.455896855"
    },
    {
      "turn": "-0.076289840",
      "x": "0.124671857",
      "y": "0.360616972"
    },
    {
      "turn": "-0.158384440",
      "x": "0.709336466",
      "y": "0.279754378"
    },
    {
      "turn": "-0.158384438",
      "x": "0.371396603",
      "y": "0.262741584"
    },
    {
      "turn": "0.158384440",
      "x": "0.190718620",
      "y": "0.113478224"
    },
    {
      "turn": "0.158384440",
      "x": "0.552074585",
      "y": "0.113478224"
    },
    {
      "turn": "0.158384440",
      "x": "0.866598347",
      "y": "0.113478224"
    }
  ],
  "radius": "0.140266798"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 16,
  "sides": 5
}

Starter answer

{
  "placements": [
    {
      "turn": "-0.076289847",
      "x": "0.635518039",
      "y": "0.874717465"
    },
    {
      "turn": "0.242615877",
      "x": "0.138643502",
      "y": "0.861566923"
    },
    {
      "turn": "-0.076289840",
      "x": "0.370275505",
      "y": "0.874717464"
    },
    {
      "turn": "-0.324919696",
      "x": "0.886521776",
      "y": "0.815029350"
    },
    {
      "turn": "0.242615877",
      "x": "0.503748883",
      "y": "0.686961985"
    },
    {
      "turn": "-0.030639622",
      "x": "0.732201065",
      "y": "0.633086977"
    },
    {
      "turn": "-0.076289840",
      "x": "0.124671857",
      "y": "0.629322530"
    },
    {
      "turn": "0.242615877",
      "x": "0.310847496",
      "y": "0.494777157"
    },
    {
      "turn": "-0.076289840",
      "x": "0.535458071",
      "y": "0.462177428"
    },
    {
      "turn": "0.324919696",
      "x": "0.886521776",
      "y": "0.455896855"
    },
    {
      "turn": "-0.076289840",
      "x": "0.124671857",
      "y": "0.360616972"
    },
    {
      "turn": "-0.158384440",
      "x": "0.709336466",
      "y": "0.279754378"
    },
    {
      "turn": "-0.158384438",
      "x": "0.371396603",
      "y": "0.262741584"
    },
    {
      "turn": "0.158384440",
      "x": "0.190718620",
      "y": "0.113478224"
    },
    {
      "turn": "0.158384440",
      "x": "0.552074585",
      "y": "0.113478224"
    },
    {
      "turn": "0.158384440",
      "x": "0.866598347",
      "y": "0.113478224"
    }
  ],
  "radius": "0.140266798"
}

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