P86 · Packing and covering · Classic · Hard

Regular pentagons in a square · n = 4

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

Instancen = 4

ObjectiveMinimize container square side

Best known, unproven3.10252Source-precision interval: 3.10252–3.10253 (values inside match)exhibited site certificate 3.102525028Compiled by Erich Friedman; Found by Jonathan Viquerat in April 2026. Source expression: s = ((3−√5)√(5−√5)+7√10+√2)/8; s = 3.10252+.

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

3.102525028

container square side

Matches the best known

Answer sourceJonathan Viquerat

Solution methodPublished reference construction

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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": "0.078701707",
      "x": "0.244296313",
      "y": "0.729195451"
    },
    {
      "turn": "-0.240078759",
      "x": "0.755703687",
      "y": "0.755703687"
    },
    {
      "turn": "-0.240081900",
      "x": "0.274230288",
      "y": "0.274252974"
    },
    {
      "turn": "0.078701707",
      "x": "0.729195451",
      "y": "0.244296313"
    }
  ],
  "radius": "0.274180160"
}

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

Instance parameters

{
  "n": 4,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.078701707",
      "x": "0.244296313",
      "y": "0.729195451"
    },
    {
      "turn": "-0.240078759",
      "x": "0.755703687",
      "y": "0.755703687"
    },
    {
      "turn": "-0.240081900",
      "x": "0.274230288",
      "y": "0.274252974"
    },
    {
      "turn": "0.078701707",
      "x": "0.729195451",
      "y": "0.244296313"
    }
  ],
  "radius": "0.274180160"
}

DISCUSSION

Discussion

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