P86 · Packing and covering · Classic · Hard

Regular pentagons in a square · n = 12

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

Instancen = 12

ObjectiveMinimize container square side

Best known, unproven5.23221Source-precision interval: 5.23221–5.23222 (values inside match)exhibited site certificate 5.232214413Compiled by Erich Friedman; Found by Timo Berthold et al in May 2026. Source expression: s = 5.23221+.

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

Current leader

5.232214413

container square side

Matches the best known

Answer sourceTimo Berthold et al

Solution methodPublished reference construction

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Record holder's solver noteNo solver note yet (expand)＋

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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": "0.154622295",
      "y": "0.868470420"
    },
    {
      "turn": "-0.158384440",
      "x": "0.595309629",
      "y": "0.868470420"
    },
    {
      "turn": "0.324919696",
      "x": "0.868470420",
      "y": "0.757613345"
    },
    {
      "turn": "0.158384440",
      "x": "0.374965962",
      "y": "0.703400459"
    },
    {
      "turn": "0.013691475",
      "x": "0.637562153",
      "y": "0.581318000"
    },
    {
      "turn": "0.000000000",
      "x": "0.131529580",
      "y": "0.551631245"
    },
    {
      "turn": "0.324919696",
      "x": "0.868470420",
      "y": "0.448368755"
    },
    {
      "turn": "-0.309849814",
      "x": "0.362437847",
      "y": "0.418682000"
    },
    {
      "turn": "-0.158384440",
      "x": "0.625034038",
      "y": "0.296599541"
    },
    {
      "turn": "0.000000000",
      "x": "0.131529580",
      "y": "0.242386655"
    },
    {
      "turn": "0.158384440",
      "x": "0.404690371",
      "y": "0.131529580"
    },
    {
      "turn": "0.158384440",
      "x": "0.845377705",
      "y": "0.131529580"
    }
  ],
  "radius": "0.162579501"
}

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

Instance parameters

{
  "n": 12,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "-0.158384440",
      "x": "0.154622295",
      "y": "0.868470420"
    },
    {
      "turn": "-0.158384440",
      "x": "0.595309629",
      "y": "0.868470420"
    },
    {
      "turn": "0.324919696",
      "x": "0.868470420",
      "y": "0.757613345"
    },
    {
      "turn": "0.158384440",
      "x": "0.374965962",
      "y": "0.703400459"
    },
    {
      "turn": "0.013691475",
      "x": "0.637562153",
      "y": "0.581318000"
    },
    {
      "turn": "0.000000000",
      "x": "0.131529580",
      "y": "0.551631245"
    },
    {
      "turn": "0.324919696",
      "x": "0.868470420",
      "y": "0.448368755"
    },
    {
      "turn": "-0.309849814",
      "x": "0.362437847",
      "y": "0.418682000"
    },
    {
      "turn": "-0.158384440",
      "x": "0.625034038",
      "y": "0.296599541"
    },
    {
      "turn": "0.000000000",
      "x": "0.131529580",
      "y": "0.242386655"
    },
    {
      "turn": "0.158384440",
      "x": "0.404690371",
      "y": "0.131529580"
    },
    {
      "turn": "0.158384440",
      "x": "0.845377705",
      "y": "0.131529580"
    }
  ],
  "radius": "0.162579501"
}

DISCUSSION

Discussion

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