P86 · Packing and covering · Classic · Hard

Regular pentagons in a square · n = 15

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

Instancen = 15

ObjectiveMinimize container square side

Best known, unproven5.90118Source-precision interval: 5.90118–5.90119 (values inside match)exhibited site certificate 5.901183982Compiled by Erich Friedman; Found by Jonathan Viquerat in August 2026. Source expression: s = 5.90118+.

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

5.901183982

container square side

Matches the best known

Answer sourceJonathan Viquerat

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": "-0.322710959",
      "x": "0.880123242",
      "y": "0.863085181"
    },
    {
      "turn": "-0.322710959",
      "x": "0.404916336",
      "y": "0.863085181"
    },
    {
      "turn": "0.324919696",
      "x": "0.144149177",
      "y": "0.862905986"
    },
    {
      "turn": "0.001999119",
      "x": "0.638415314",
      "y": "0.798345648"
    },
    {
      "turn": "0.000000000",
      "x": "0.116619134",
      "y": "0.608719704"
    },
    {
      "turn": "0.001999119",
      "x": "0.377389427",
      "y": "0.608457107"
    },
    {
      "turn": "0.034313024",
      "x": "0.856189861",
      "y": "0.604137971"
    },
    {
      "turn": "-0.287402429",
      "x": "0.625730780",
      "y": "0.555684994"
    },
    {
      "turn": "0.324919696",
      "x": "0.255147777",
      "y": "0.402579381"
    },
    {
      "turn": "0.310301669",
      "x": "0.809319273",
      "y": "0.367677723"
    },
    {
      "turn": "0.000000000",
      "x": "0.488386045",
      "y": "0.347768834"
    },
    {
      "turn": "0.000000000",
      "x": "0.116619134",
      "y": "0.196439058"
    },
    {
      "turn": "-0.013279179",
      "x": "0.855901651",
      "y": "0.138228495"
    },
    {
      "turn": "-0.324919696",
      "x": "0.361800790",
      "y": "0.137094014"
    },
    {
      "turn": "0.310301669",
      "x": "0.622518273",
      "y": "0.135862852"
    }
  ],
  "radius": "0.144149176"
}

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

Instance parameters

{
  "n": 15,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "-0.322710959",
      "x": "0.880123242",
      "y": "0.863085181"
    },
    {
      "turn": "-0.322710959",
      "x": "0.404916336",
      "y": "0.863085181"
    },
    {
      "turn": "0.324919696",
      "x": "0.144149177",
      "y": "0.862905986"
    },
    {
      "turn": "0.001999119",
      "x": "0.638415314",
      "y": "0.798345648"
    },
    {
      "turn": "0.000000000",
      "x": "0.116619134",
      "y": "0.608719704"
    },
    {
      "turn": "0.001999119",
      "x": "0.377389427",
      "y": "0.608457107"
    },
    {
      "turn": "0.034313024",
      "x": "0.856189861",
      "y": "0.604137971"
    },
    {
      "turn": "-0.287402429",
      "x": "0.625730780",
      "y": "0.555684994"
    },
    {
      "turn": "0.324919696",
      "x": "0.255147777",
      "y": "0.402579381"
    },
    {
      "turn": "0.310301669",
      "x": "0.809319273",
      "y": "0.367677723"
    },
    {
      "turn": "0.000000000",
      "x": "0.488386045",
      "y": "0.347768834"
    },
    {
      "turn": "0.000000000",
      "x": "0.116619134",
      "y": "0.196439058"
    },
    {
      "turn": "-0.013279179",
      "x": "0.855901651",
      "y": "0.138228495"
    },
    {
      "turn": "-0.324919696",
      "x": "0.361800790",
      "y": "0.137094014"
    },
    {
      "turn": "0.310301669",
      "x": "0.622518273",
      "y": "0.135862852"
    }
  ],
  "radius": "0.144149176"
}

DISCUSSION

Discussion

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