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P86 · Packing and covering · Classic · Hard

Regular pentagons in a square · n = 24

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

Instancen = 24
ObjectiveMinimize container square side

Record comparison

container square side · Lower is better
RecordValue / intervalAuthor / holderSource
External best known[7.3369, 7.33691)Jake LoydErich 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.
Open full editor ↗

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 = 24
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": "-0.158384440",
      "x": "0.296181314",
      "y": "0.906201450"
    },
    {
      "turn": "-0.158384440",
      "x": "0.524453858",
      "y": "0.906201450"
    },
    {
      "turn": "-0.000000000",
      "x": "0.728519217",
      "y": "0.889733191"
    },
    {
      "turn": "-0.000000000",
      "x": "0.093798550",
      "y": "0.837580703"
    },
    {
      "turn": "0.324919696",
      "x": "0.906201450",
      "y": "0.815132555"
    },
    {
      "turn": "0.158384440",
      "x": "0.410317586",
      "y": "0.757243537"
    },
    {
      "turn": "-0.324919696",
      "x": "0.217864562",
      "y": "0.689183669"
    },
    {
      "turn": "0.000000000",
      "x": "0.770447318",
      "y": "0.682822840"
    },
    {
      "turn": "0.324919696",
      "x": "0.582850218",
      "y": "0.647537607"
    },
    {
      "turn": "-0.158384440",
      "x": "0.393591577",
      "y": "0.569646437"
    },
    {
      "turn": "-0.324919696",
      "x": "0.906201450",
      "y": "0.550513124"
    },
    {
      "turn": "-0.000000000",
      "x": "0.093798550",
      "y": "0.540786635"
    },
    {
      "turn": "-0.000000000",
      "x": "0.731881517",
      "y": "0.471284678"
    },
    {
      "turn": "0.158384440",
      "x": "0.527816159",
      "y": "0.435283536"
    },
    {
      "turn": "0.158384440",
      "x": "0.265923679",
      "y": "0.430519826"
    },
    {
      "turn": "0.222391217",
      "x": "0.894986567",
      "y": "0.328544586"
    },
    {
      "turn": "-0.000000000",
      "x": "0.093798550",
      "y": "0.320253018"
    },
    {
      "turn": "-0.095619111",
      "x": "0.719689372",
      "y": "0.260538577"
    },
    {
      "turn": "-0.158384440",
      "x": "0.504497605",
      "y": "0.247686436"
    },
    {
      "turn": "-0.158384440",
      "x": "0.283239431",
      "y": "0.242922726"
    },
    {
      "turn": "0.159906789",
      "x": "0.149819144",
      "y": "0.107223119"
    },
    {
      "turn": "-0.195247848",
      "x": "0.869370224",
      "y": "0.115645879"
    },
    {
      "turn": "0.158384440",
      "x": "0.397146860",
      "y": "0.093798550"
    },
    {
      "turn": "0.158384440",
      "x": "0.617680478",
      "y": "0.093798550"
    }
  ],
  "radius": "0.115941383"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 24,
  "sides": 5
}

Starter answer

{
  "placements": [
    {
      "turn": "-0.158384440",
      "x": "0.296181314",
      "y": "0.906201450"
    },
    {
      "turn": "-0.158384440",
      "x": "0.524453858",
      "y": "0.906201450"
    },
    {
      "turn": "-0.000000000",
      "x": "0.728519217",
      "y": "0.889733191"
    },
    {
      "turn": "-0.000000000",
      "x": "0.093798550",
      "y": "0.837580703"
    },
    {
      "turn": "0.324919696",
      "x": "0.906201450",
      "y": "0.815132555"
    },
    {
      "turn": "0.158384440",
      "x": "0.410317586",
      "y": "0.757243537"
    },
    {
      "turn": "-0.324919696",
      "x": "0.217864562",
      "y": "0.689183669"
    },
    {
      "turn": "0.000000000",
      "x": "0.770447318",
      "y": "0.682822840"
    },
    {
      "turn": "0.324919696",
      "x": "0.582850218",
      "y": "0.647537607"
    },
    {
      "turn": "-0.158384440",
      "x": "0.393591577",
      "y": "0.569646437"
    },
    {
      "turn": "-0.324919696",
      "x": "0.906201450",
      "y": "0.550513124"
    },
    {
      "turn": "-0.000000000",
      "x": "0.093798550",
      "y": "0.540786635"
    },
    {
      "turn": "-0.000000000",
      "x": "0.731881517",
      "y": "0.471284678"
    },
    {
      "turn": "0.158384440",
      "x": "0.527816159",
      "y": "0.435283536"
    },
    {
      "turn": "0.158384440",
      "x": "0.265923679",
      "y": "0.430519826"
    },
    {
      "turn": "0.222391217",
      "x": "0.894986567",
      "y": "0.328544586"
    },
    {
      "turn": "-0.000000000",
      "x": "0.093798550",
      "y": "0.320253018"
    },
    {
      "turn": "-0.095619111",
      "x": "0.719689372",
      "y": "0.260538577"
    },
    {
      "turn": "-0.158384440",
      "x": "0.504497605",
      "y": "0.247686436"
    },
    {
      "turn": "-0.158384440",
      "x": "0.283239431",
      "y": "0.242922726"
    },
    {
      "turn": "0.159906789",
      "x": "0.149819144",
      "y": "0.107223119"
    },
    {
      "turn": "-0.195247848",
      "x": "0.869370224",
      "y": "0.115645879"
    },
    {
      "turn": "0.158384440",
      "x": "0.397146860",
      "y": "0.093798550"
    },
    {
      "turn": "0.158384440",
      "x": "0.617680478",
      "y": "0.093798550"
    }
  ],
  "radius": "0.115941383"
}

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