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P86 · Packing and covering · Classic · Hard

Regular pentagons in a square · n = 15

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

Instancen = 15
ObjectiveMinimize container square side

Record comparison

container square side · Lower is better
RecordValue / intervalAuthor / holderSource
External best known[5.90118, 5.90119)Jonathan ViqueratErich 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 = 15
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": "-0.322710959",
      "x": "0.880123242",
      "y": "0.863085181"
    },
    {
      "turn": "-0.322710959",
      "x": "0.404916336",
      "y": "0.863085181"
    },
    {
      "turn": "0.324919696",
      "x": "0.144149177",
      "y": "0.862905986"
    },
    {
      "turn": "0.001999119",
      "x": "0.638415314",
      "y": "0.798345648"
    },
    {
      "turn": "0.000000000",
      "x": "0.116619134",
      "y": "0.608719704"
    },
    {
      "turn": "0.001999119",
      "x": "0.377389427",
      "y": "0.608457107"
    },
    {
      "turn": "0.034313024",
      "x": "0.856189861",
      "y": "0.604137971"
    },
    {
      "turn": "-0.287402429",
      "x": "0.625730780",
      "y": "0.555684994"
    },
    {
      "turn": "0.324919696",
      "x": "0.255147777",
      "y": "0.402579381"
    },
    {
      "turn": "0.310301669",
      "x": "0.809319273",
      "y": "0.367677723"
    },
    {
      "turn": "0.000000000",
      "x": "0.488386045",
      "y": "0.347768834"
    },
    {
      "turn": "0.000000000",
      "x": "0.116619134",
      "y": "0.196439058"
    },
    {
      "turn": "-0.013279179",
      "x": "0.855901651",
      "y": "0.138228495"
    },
    {
      "turn": "-0.324919696",
      "x": "0.361800790",
      "y": "0.137094014"
    },
    {
      "turn": "0.310301669",
      "x": "0.622518273",
      "y": "0.135862852"
    }
  ],
  "radius": "0.144149176"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 15,
  "sides": 5
}

Starter answer

{
  "placements": [
    {
      "turn": "-0.322710959",
      "x": "0.880123242",
      "y": "0.863085181"
    },
    {
      "turn": "-0.322710959",
      "x": "0.404916336",
      "y": "0.863085181"
    },
    {
      "turn": "0.324919696",
      "x": "0.144149177",
      "y": "0.862905986"
    },
    {
      "turn": "0.001999119",
      "x": "0.638415314",
      "y": "0.798345648"
    },
    {
      "turn": "0.000000000",
      "x": "0.116619134",
      "y": "0.608719704"
    },
    {
      "turn": "0.001999119",
      "x": "0.377389427",
      "y": "0.608457107"
    },
    {
      "turn": "0.034313024",
      "x": "0.856189861",
      "y": "0.604137971"
    },
    {
      "turn": "-0.287402429",
      "x": "0.625730780",
      "y": "0.555684994"
    },
    {
      "turn": "0.324919696",
      "x": "0.255147777",
      "y": "0.402579381"
    },
    {
      "turn": "0.310301669",
      "x": "0.809319273",
      "y": "0.367677723"
    },
    {
      "turn": "0.000000000",
      "x": "0.488386045",
      "y": "0.347768834"
    },
    {
      "turn": "0.000000000",
      "x": "0.116619134",
      "y": "0.196439058"
    },
    {
      "turn": "-0.013279179",
      "x": "0.855901651",
      "y": "0.138228495"
    },
    {
      "turn": "-0.324919696",
      "x": "0.361800790",
      "y": "0.137094014"
    },
    {
      "turn": "0.310301669",
      "x": "0.622518273",
      "y": "0.135862852"
    }
  ],
  "radius": "0.144149176"
}

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