P86 · Packing and covering · Classic · Hard

Regular pentagons in a square · n = 9

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

Instancen = 9

ObjectiveMinimize container square side

Best known, unproven4.60033Source-precision interval: 4.60033–4.60034 (values inside match)exhibited site certificate 4.600330011Compiled by Erich Friedman; Found by Jonathan Viquerat in April 2026. Source expression: s = 4.60033+.

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.

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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

4.600330011

container square side

Matches the best known

Answer sourceJonathan Viquerat

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": "0.078655778",
      "x": "0.164749090",
      "y": "0.817368368"
    },
    {
      "turn": "-0.240127037",
      "x": "0.509661591",
      "y": "0.835235572"
    },
    {
      "turn": "0.078833508",
      "x": "0.815916220",
      "y": "0.815920672"
    },
    {
      "turn": "-0.240127035",
      "x": "0.182636904",
      "y": "0.509651982"
    },
    {
      "turn": "-0.240134156",
      "x": "0.835252029",
      "y": "0.509651366"
    },
    {
      "turn": "0.078655779",
      "x": "0.488845748",
      "y": "0.488856347"
    },
    {
      "turn": "0.078655779",
      "x": "0.164749252",
      "y": "0.164764413"
    },
    {
      "turn": "-0.240127037",
      "x": "0.509661756",
      "y": "0.182631623"
    },
    {
      "turn": "0.078649005",
      "x": "0.817362697",
      "y": "0.164765548"
    }
  ],
  "radius": "0.184910823"
}

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

Instance parameters

{
  "n": 9,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.078655778",
      "x": "0.164749090",
      "y": "0.817368368"
    },
    {
      "turn": "-0.240127037",
      "x": "0.509661591",
      "y": "0.835235572"
    },
    {
      "turn": "0.078833508",
      "x": "0.815916220",
      "y": "0.815920672"
    },
    {
      "turn": "-0.240127035",
      "x": "0.182636904",
      "y": "0.509651982"
    },
    {
      "turn": "-0.240134156",
      "x": "0.835252029",
      "y": "0.509651366"
    },
    {
      "turn": "0.078655779",
      "x": "0.488845748",
      "y": "0.488856347"
    },
    {
      "turn": "0.078655779",
      "x": "0.164749252",
      "y": "0.164764413"
    },
    {
      "turn": "-0.240127037",
      "x": "0.509661756",
      "y": "0.182631623"
    },
    {
      "turn": "0.078649005",
      "x": "0.817362697",
      "y": "0.164765548"
    }
  ],
  "radius": "0.184910823"
}

DISCUSSION

Discussion

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