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P86 · Packing and covering · Classic · Hard

Regular pentagons in a square · n = 22

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

Instancen = 22
ObjectiveMinimize container square side
Best known, unproven7.04445Source-precision interval: 7.04445–7.04446 (values inside match)exhibited site certificate 7.044451306Compiled by Erich Friedman; Found by Haowei Lin in July 2026. Source expression: s = 7.04445+.

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 = 22, 7.044451306
Current leader

7.044451306

container square side

Matches the best known
Answer sourceHaowei Lin
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": "0.594926949",
      "y": "0.902307372"
    },
    {
      "turn": "-0.158384440",
      "x": "0.365237804",
      "y": "0.902307372"
    },
    {
      "turn": "-0.170247844",
      "x": "0.825113038",
      "y": "0.900688478"
    },
    {
      "turn": "-0.003948438",
      "x": "0.105078212",
      "y": "0.873247276"
    },
    {
      "turn": "0.158384440",
      "x": "0.690643273",
      "y": "0.730339892"
    },
    {
      "turn": "0.215972054",
      "x": "0.279776261",
      "y": "0.717768101"
    },
    {
      "turn": "-0.087424094",
      "x": "0.480082377",
      "y": "0.717186315"
    },
    {
      "turn": "-0.324919696",
      "x": "0.902307372",
      "y": "0.693990174"
    },
    {
      "turn": "0.000000000",
      "x": "0.097692628",
      "y": "0.616409814"
    },
    {
      "turn": "-0.148830416",
      "x": "0.689552407",
      "y": "0.533646757"
    },
    {
      "turn": "-0.321484634",
      "x": "0.477686249",
      "y": "0.501206192"
    },
    {
      "turn": "-0.299519441",
      "x": "0.259121478",
      "y": "0.500778967"
    },
    {
      "turn": "-0.317923412",
      "x": "0.901415089",
      "y": "0.465512915"
    },
    {
      "turn": "0.023147542",
      "x": "0.100872161",
      "y": "0.385823565"
    },
    {
      "turn": "0.006341237",
      "x": "0.757285939",
      "y": "0.331957129"
    },
    {
      "turn": "0.023147542",
      "x": "0.320678301",
      "y": "0.315177460"
    },
    {
      "turn": "0.003110183",
      "x": "0.539029718",
      "y": "0.315700352"
    },
    {
      "turn": "0.324919696",
      "x": "0.902307372",
      "y": "0.197634712"
    },
    {
      "turn": "0.000000000",
      "x": "0.097692628",
      "y": "0.160304224"
    },
    {
      "turn": "-0.299519441",
      "x": "0.306535217",
      "y": "0.116448157"
    },
    {
      "turn": "-0.321484634",
      "x": "0.525183406",
      "y": "0.115074463"
    },
    {
      "turn": "0.003110183",
      "x": "0.720575331",
      "y": "0.114610238"
    }
  ],
  "radius": "0.120754729"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 22,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "-0.158384440",
      "x": "0.594926949",
      "y": "0.902307372"
    },
    {
      "turn": "-0.158384440",
      "x": "0.365237804",
      "y": "0.902307372"
    },
    {
      "turn": "-0.170247844",
      "x": "0.825113038",
      "y": "0.900688478"
    },
    {
      "turn": "-0.003948438",
      "x": "0.105078212",
      "y": "0.873247276"
    },
    {
      "turn": "0.158384440",
      "x": "0.690643273",
      "y": "0.730339892"
    },
    {
      "turn": "0.215972054",
      "x": "0.279776261",
      "y": "0.717768101"
    },
    {
      "turn": "-0.087424094",
      "x": "0.480082377",
      "y": "0.717186315"
    },
    {
      "turn": "-0.324919696",
      "x": "0.902307372",
      "y": "0.693990174"
    },
    {
      "turn": "0.000000000",
      "x": "0.097692628",
      "y": "0.616409814"
    },
    {
      "turn": "-0.148830416",
      "x": "0.689552407",
      "y": "0.533646757"
    },
    {
      "turn": "-0.321484634",
      "x": "0.477686249",
      "y": "0.501206192"
    },
    {
      "turn": "-0.299519441",
      "x": "0.259121478",
      "y": "0.500778967"
    },
    {
      "turn": "-0.317923412",
      "x": "0.901415089",
      "y": "0.465512915"
    },
    {
      "turn": "0.023147542",
      "x": "0.100872161",
      "y": "0.385823565"
    },
    {
      "turn": "0.006341237",
      "x": "0.757285939",
      "y": "0.331957129"
    },
    {
      "turn": "0.023147542",
      "x": "0.320678301",
      "y": "0.315177460"
    },
    {
      "turn": "0.003110183",
      "x": "0.539029718",
      "y": "0.315700352"
    },
    {
      "turn": "0.324919696",
      "x": "0.902307372",
      "y": "0.197634712"
    },
    {
      "turn": "0.000000000",
      "x": "0.097692628",
      "y": "0.160304224"
    },
    {
      "turn": "-0.299519441",
      "x": "0.306535217",
      "y": "0.116448157"
    },
    {
      "turn": "-0.321484634",
      "x": "0.525183406",
      "y": "0.115074463"
    },
    {
      "turn": "0.003110183",
      "x": "0.720575331",
      "y": "0.114610238"
    }
  ],
  "radius": "0.120754729"
}

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

DISCUSSION

Discussion

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