P86 · Packing and covering · Classic · Hard

Regular pentagons in a square · n = 17

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

Instancen = 17

ObjectiveMinimize container square side

Record comparison

container square side · Lower is better

Record	Value / interval	Author / holder	Source
External best known	[6.23761, 6.23762)	Jonathan Viquerat	Erich Friedman ↗
Site record	No 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

LAYOUT & HISTORY

Starter layout

Inspect the layout, solver notes and record history.

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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": "-0.158384440",
      "x": "0.328446781",
      "y": "0.889670863"
    },
    {
      "turn": "-0.158384440",
      "x": "0.661897877",
      "y": "0.889670863"
    },
    {
      "turn": "-0.324919696",
      "x": "0.889670863",
      "y": "0.794812070"
    },
    {
      "turn": "-0.000000000",
      "x": "0.110329137",
      "y": "0.781522632"
    },
    {
      "turn": "0.158384440",
      "x": "0.495172329",
      "y": "0.738055438"
    },
    {
      "turn": "-0.000000000",
      "x": "0.719928920",
      "y": "0.653035533"
    },
    {
      "turn": "0.324919696",
      "x": "0.276383067",
      "y": "0.634669982"
    },
    {
      "turn": "-0.324919696",
      "x": "0.889670863",
      "y": "0.511258996"
    },
    {
      "turn": "-0.158384440",
      "x": "0.494424188",
      "y": "0.517397164"
    },
    {
      "turn": "0.000000000",
      "x": "0.110329137",
      "y": "0.487817331"
    },
    {
      "turn": "-0.000000000",
      "x": "0.729302476",
      "y": "0.356580867"
    },
    {
      "turn": "-0.324919696",
      "x": "0.274385536",
      "y": "0.338215315"
    },
    {
      "turn": "-0.158384440",
      "x": "0.504827671",
      "y": "0.270693714"
    },
    {
      "turn": "0.324919696",
      "x": "0.889670863",
      "y": "0.201902737"
    },
    {
      "turn": "0.000000000",
      "x": "0.110329137",
      "y": "0.188613299"
    },
    {
      "turn": "0.158384440",
      "x": "0.350144297",
      "y": "0.110329137"
    },
    {
      "turn": "0.158384440",
      "x": "0.659511045",
      "y": "0.110329137"
    }
  ],
  "radius": "0.136374312"
}

Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 17,
  "sides": 5
}

Starter answer

{
  "placements": [
    {
      "turn": "-0.158384440",
      "x": "0.328446781",
      "y": "0.889670863"
    },
    {
      "turn": "-0.158384440",
      "x": "0.661897877",
      "y": "0.889670863"
    },
    {
      "turn": "-0.324919696",
      "x": "0.889670863",
      "y": "0.794812070"
    },
    {
      "turn": "-0.000000000",
      "x": "0.110329137",
      "y": "0.781522632"
    },
    {
      "turn": "0.158384440",
      "x": "0.495172329",
      "y": "0.738055438"
    },
    {
      "turn": "-0.000000000",
      "x": "0.719928920",
      "y": "0.653035533"
    },
    {
      "turn": "0.324919696",
      "x": "0.276383067",
      "y": "0.634669982"
    },
    {
      "turn": "-0.324919696",
      "x": "0.889670863",
      "y": "0.511258996"
    },
    {
      "turn": "-0.158384440",
      "x": "0.494424188",
      "y": "0.517397164"
    },
    {
      "turn": "0.000000000",
      "x": "0.110329137",
      "y": "0.487817331"
    },
    {
      "turn": "-0.000000000",
      "x": "0.729302476",
      "y": "0.356580867"
    },
    {
      "turn": "-0.324919696",
      "x": "0.274385536",
      "y": "0.338215315"
    },
    {
      "turn": "-0.158384440",
      "x": "0.504827671",
      "y": "0.270693714"
    },
    {
      "turn": "0.324919696",
      "x": "0.889670863",
      "y": "0.201902737"
    },
    {
      "turn": "0.000000000",
      "x": "0.110329137",
      "y": "0.188613299"
    },
    {
      "turn": "0.158384440",
      "x": "0.350144297",
      "y": "0.110329137"
    },
    {
      "turn": "0.158384440",
      "x": "0.659511045",
      "y": "0.110329137"
    }
  ],
  "radius": "0.136374312"
}

DISCUSSION

Discussion

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外部记录更新NUE_13#311 replies39d ago