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P86 · Packing and covering · Classic · Hard

Regular pentagons in a square · n = 16

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

Instancen = 16
ObjectiveMinimize container square side
Best known, unproven6.06452Source-precision interval: 6.06452–6.06453 (values inside match)exhibited site certificate 6.064520047Compiled by Erich Friedman; Found by Timo Berthold et al in April 2026. Source expression: s = 6.06452+.

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 = 16, 6.064520047
Current leader

6.064520047

container square side

Matches the best known
Answer sourceTimo Berthold et al
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": "-0.076289847",
      "x": "0.635518039",
      "y": "0.874717465"
    },
    {
      "turn": "0.242615877",
      "x": "0.138643502",
      "y": "0.861566923"
    },
    {
      "turn": "-0.076289840",
      "x": "0.370275505",
      "y": "0.874717464"
    },
    {
      "turn": "-0.324919696",
      "x": "0.886521776",
      "y": "0.815029350"
    },
    {
      "turn": "0.242615877",
      "x": "0.503748883",
      "y": "0.686961985"
    },
    {
      "turn": "-0.030639622",
      "x": "0.732201065",
      "y": "0.633086977"
    },
    {
      "turn": "-0.076289840",
      "x": "0.124671857",
      "y": "0.629322530"
    },
    {
      "turn": "0.242615877",
      "x": "0.310847496",
      "y": "0.494777157"
    },
    {
      "turn": "-0.076289840",
      "x": "0.535458071",
      "y": "0.462177428"
    },
    {
      "turn": "0.324919696",
      "x": "0.886521776",
      "y": "0.455896855"
    },
    {
      "turn": "-0.076289840",
      "x": "0.124671857",
      "y": "0.360616972"
    },
    {
      "turn": "-0.158384440",
      "x": "0.709336466",
      "y": "0.279754378"
    },
    {
      "turn": "-0.158384438",
      "x": "0.371396603",
      "y": "0.262741584"
    },
    {
      "turn": "0.158384440",
      "x": "0.190718620",
      "y": "0.113478224"
    },
    {
      "turn": "0.158384440",
      "x": "0.552074585",
      "y": "0.113478224"
    },
    {
      "turn": "0.158384440",
      "x": "0.866598347",
      "y": "0.113478224"
    }
  ],
  "radius": "0.140266798"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 16,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "-0.076289847",
      "x": "0.635518039",
      "y": "0.874717465"
    },
    {
      "turn": "0.242615877",
      "x": "0.138643502",
      "y": "0.861566923"
    },
    {
      "turn": "-0.076289840",
      "x": "0.370275505",
      "y": "0.874717464"
    },
    {
      "turn": "-0.324919696",
      "x": "0.886521776",
      "y": "0.815029350"
    },
    {
      "turn": "0.242615877",
      "x": "0.503748883",
      "y": "0.686961985"
    },
    {
      "turn": "-0.030639622",
      "x": "0.732201065",
      "y": "0.633086977"
    },
    {
      "turn": "-0.076289840",
      "x": "0.124671857",
      "y": "0.629322530"
    },
    {
      "turn": "0.242615877",
      "x": "0.310847496",
      "y": "0.494777157"
    },
    {
      "turn": "-0.076289840",
      "x": "0.535458071",
      "y": "0.462177428"
    },
    {
      "turn": "0.324919696",
      "x": "0.886521776",
      "y": "0.455896855"
    },
    {
      "turn": "-0.076289840",
      "x": "0.124671857",
      "y": "0.360616972"
    },
    {
      "turn": "-0.158384440",
      "x": "0.709336466",
      "y": "0.279754378"
    },
    {
      "turn": "-0.158384438",
      "x": "0.371396603",
      "y": "0.262741584"
    },
    {
      "turn": "0.158384440",
      "x": "0.190718620",
      "y": "0.113478224"
    },
    {
      "turn": "0.158384440",
      "x": "0.552074585",
      "y": "0.113478224"
    },
    {
      "turn": "0.158384440",
      "x": "0.866598347",
      "y": "0.113478224"
    }
  ],
  "radius": "0.140266798"
}

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

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