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P86 · Packing and covering · Classic · Hard

Regular pentagons in a square · n = 30

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

Instancen = 30
ObjectiveMinimize container square side
Best known, unproven8.10222Source-precision interval: 8.10222–8.10223 (values inside match)exhibited site certificate 8.102220102Compiled by Erich Friedman; Found by Jonathan Viquerat in August 2026. Source expression: s = 8.10222+.

Formal definition

  • ContainerThe editor and certificate normalize to the unit square [0,1]². This is a coordinate convention only; scores use the literature's unit-piece scale. Boundary contact is allowed.
  • SubmissionSubmit {radius, placements:[{x,y,turn},…]}, with decimal strings of at most nine places. x,y are centres; circles require turn=0, polygons use turn=tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4.
  • ConstraintsExactly n congruent shapes lie wholly in the container with pairwise disjoint interiors. Polygon rotations are independent, not limited to quarter turns.
  • PrecisionInput decimals are exact rationals; regular-polygon vertices are algebraic, not rounded templates. Feasibility has no floating-point tolerance. A finite-decimal certificate is not a continuous optimality proof.
  • Objectivecontainer square side = 1/(2 sin(pi/5) r), smaller is better. Pages, leaderboards and literature share these units; conversion is automatic.
Open the full editor ↗
VERIFIED CONSTRUCTIONVERIFIED CONSTRUCTION

Getting a feel for it

Source and certificate

External targets are converted with their printed precision; decimal coordinate certificates are separate. Reconstruction loss never lowers the literature target, and a source construction is not automatically an optimality proof.

Source
The record-holding arrangement for Regular pentagons in a square n = 30, 8.102220102
Current leader

8.102220102

container square side

Matches the best known
Answer sourceJonathan Viquerat
Solution methodPublished reference construction
Challenge this record ↗Submit a proof / idea ↓Share a proof or idea in the discussion. Accepted contributions can earn proof points.
Record holder's solver noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

ANSWER FORMAT

How to write your answer

the unit square [0,1]²

Submit {radius, placements:[{x,y,turn},…]}, with decimal strings of at most nine places. x,y are centres; circles require turn=0, polygons use turn=tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4.

The current leader's answer

{
  "placements": [
    {
      "turn": "-0.158384440",
      "x": "0.145249409",
      "y": "0.915061433"
    },
    {
      "turn": "-0.008321199",
      "x": "0.330352007",
      "y": "0.900702455"
    },
    {
      "turn": "0.324919696",
      "x": "0.915061433",
      "y": "0.900148725"
    },
    {
      "turn": "-0.118401546",
      "x": "0.524891975",
      "y": "0.908160186"
    },
    {
      "turn": "-0.324919696",
      "x": "0.725133022",
      "y": "0.900148725"
    },
    {
      "turn": "0.198867519",
      "x": "0.445294897",
      "y": "0.757699641"
    },
    {
      "turn": "-0.000000000",
      "x": "0.813901042",
      "y": "0.750371813"
    },
    {
      "turn": "0.226648004",
      "x": "0.103245261",
      "y": "0.736225592"
    },
    {
      "turn": "-0.029968489",
      "x": "0.624161029",
      "y": "0.744084710"
    },
    {
      "turn": "-0.091531121",
      "x": "0.273189733",
      "y": "0.721041305"
    },
    {
      "turn": "0.260749998",
      "x": "0.724429250",
      "y": "0.597206396"
    },
    {
      "turn": "0.324919696",
      "x": "0.915061433",
      "y": "0.598109080"
    },
    {
      "turn": "0.226648004",
      "x": "0.372649795",
      "y": "0.581209499"
    },
    {
      "turn": "-0.091531121",
      "x": "0.543779907",
      "y": "0.572447648"
    },
    {
      "turn": "-0.160110396",
      "x": "0.172241524",
      "y": "0.560810112"
    },
    {
      "turn": "-0.011204754",
      "x": "0.816923018",
      "y": "0.441856579"
    },
    {
      "turn": "0.312576960",
      "x": "0.646596965",
      "y": "0.423730037"
    },
    {
      "turn": "0.167374907",
      "x": "0.271238181",
      "y": "0.421192812"
    },
    {
      "turn": "-0.232987043",
      "x": "0.457924610",
      "y": "0.407444793"
    },
    {
      "turn": "-0.000000000",
      "x": "0.084938567",
      "y": "0.395851459"
    },
    {
      "turn": "0.158874204",
      "x": "0.900179770",
      "y": "0.274867424"
    },
    {
      "turn": "0.085462931",
      "x": "0.551622702",
      "y": "0.264793862"
    },
    {
      "turn": "-0.000000000",
      "x": "0.357267028",
      "y": "0.255459349"
    },
    {
      "turn": "-0.324919696",
      "x": "0.187389893",
      "y": "0.247851367"
    },
    {
      "turn": "-0.157894751",
      "x": "0.731717872",
      "y": "0.243536798"
    },
    {
      "turn": "0.000000000",
      "x": "0.084938567",
      "y": "0.099851275"
    },
    {
      "turn": "-0.232987043",
      "x": "0.477235664",
      "y": "0.103908489"
    },
    {
      "turn": "-0.157894751",
      "x": "0.899743919",
      "y": "0.104989795"
    },
    {
      "turn": "0.158384440",
      "x": "0.284083538",
      "y": "0.084938567"
    },
    {
      "turn": "0.158384685",
      "x": "0.660955469",
      "y": "0.084938597"
    }
  ],
  "radius": "0.104989842"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 30,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "-0.158384440",
      "x": "0.145249409",
      "y": "0.915061433"
    },
    {
      "turn": "-0.008321199",
      "x": "0.330352007",
      "y": "0.900702455"
    },
    {
      "turn": "0.324919696",
      "x": "0.915061433",
      "y": "0.900148725"
    },
    {
      "turn": "-0.118401546",
      "x": "0.524891975",
      "y": "0.908160186"
    },
    {
      "turn": "-0.324919696",
      "x": "0.725133022",
      "y": "0.900148725"
    },
    {
      "turn": "0.198867519",
      "x": "0.445294897",
      "y": "0.757699641"
    },
    {
      "turn": "-0.000000000",
      "x": "0.813901042",
      "y": "0.750371813"
    },
    {
      "turn": "0.226648004",
      "x": "0.103245261",
      "y": "0.736225592"
    },
    {
      "turn": "-0.029968489",
      "x": "0.624161029",
      "y": "0.744084710"
    },
    {
      "turn": "-0.091531121",
      "x": "0.273189733",
      "y": "0.721041305"
    },
    {
      "turn": "0.260749998",
      "x": "0.724429250",
      "y": "0.597206396"
    },
    {
      "turn": "0.324919696",
      "x": "0.915061433",
      "y": "0.598109080"
    },
    {
      "turn": "0.226648004",
      "x": "0.372649795",
      "y": "0.581209499"
    },
    {
      "turn": "-0.091531121",
      "x": "0.543779907",
      "y": "0.572447648"
    },
    {
      "turn": "-0.160110396",
      "x": "0.172241524",
      "y": "0.560810112"
    },
    {
      "turn": "-0.011204754",
      "x": "0.816923018",
      "y": "0.441856579"
    },
    {
      "turn": "0.312576960",
      "x": "0.646596965",
      "y": "0.423730037"
    },
    {
      "turn": "0.167374907",
      "x": "0.271238181",
      "y": "0.421192812"
    },
    {
      "turn": "-0.232987043",
      "x": "0.457924610",
      "y": "0.407444793"
    },
    {
      "turn": "-0.000000000",
      "x": "0.084938567",
      "y": "0.395851459"
    },
    {
      "turn": "0.158874204",
      "x": "0.900179770",
      "y": "0.274867424"
    },
    {
      "turn": "0.085462931",
      "x": "0.551622702",
      "y": "0.264793862"
    },
    {
      "turn": "-0.000000000",
      "x": "0.357267028",
      "y": "0.255459349"
    },
    {
      "turn": "-0.324919696",
      "x": "0.187389893",
      "y": "0.247851367"
    },
    {
      "turn": "-0.157894751",
      "x": "0.731717872",
      "y": "0.243536798"
    },
    {
      "turn": "0.000000000",
      "x": "0.084938567",
      "y": "0.099851275"
    },
    {
      "turn": "-0.232987043",
      "x": "0.477235664",
      "y": "0.103908489"
    },
    {
      "turn": "-0.157894751",
      "x": "0.899743919",
      "y": "0.104989795"
    },
    {
      "turn": "0.158384440",
      "x": "0.284083538",
      "y": "0.084938567"
    },
    {
      "turn": "0.158384685",
      "x": "0.660955469",
      "y": "0.084938597"
    }
  ],
  "radius": "0.104989842"
}

Submit {radius, placements:[{x,y,turn},…]}, with decimal strings of at most nine places. x,y are centres; circles require turn=0, polygons use turn=tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4. · Verifier v1.0.0

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