P86 · Packing and covering · Classic · Hard

Regular pentagons in a square · n = 10

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

Instancen = 10

ObjectiveMinimize container square side

Best known, unproven4.90581Source-precision interval: 4.90581–4.90582 (values inside match)Compiled by Erich Friedman; Found by Jonathan Viquerat in June 2026. Source expression: s = 4.90581+.

The external best-known value is cited above; its layout is not reproduced here. The starting layout is an ordinary site baseline, not the external record.

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

5.316567553

container square side

Record holderXSsMC

Solution methodHuman

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.

Record history (1 changes)

XSsMCHuman
9.357158883→5.316567553
2026-09-14

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": "0.000000000",
      "x": "0.255305326",
      "y": "0.158639193"
    },
    {
      "turn": "-0.324919696",
      "x": "0.165797017",
      "y": "0.409554940"
    },
    {
      "turn": "0.000000000",
      "x": "0.129442724",
      "y": "0.699450090"
    },
    {
      "turn": "-0.324919696",
      "x": "0.351549862",
      "y": "0.847830953"
    },
    {
      "turn": "0.000000000",
      "x": "0.616232451",
      "y": "0.845620609"
    },
    {
      "turn": "0.000000000",
      "x": "0.459411461",
      "y": "0.517774346"
    },
    {
      "turn": "0.000000000",
      "x": "0.546197369",
      "y": "0.221886609"
    },
    {
      "turn": "-0.324919696",
      "x": "0.863310944",
      "y": "0.733487348"
    },
    {
      "turn": "0.000000000",
      "x": "0.839999996",
      "y": "0.152169047"
    },
    {
      "turn": "0.000000000",
      "x": "0.750309572",
      "y": "0.437606168"
    }
  ],
  "radius": "0.160000000"
}

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

Instance parameters

{
  "n": 10,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "0.255305326",
      "y": "0.158639193"
    },
    {
      "turn": "-0.324919696",
      "x": "0.165797017",
      "y": "0.409554940"
    },
    {
      "turn": "0.000000000",
      "x": "0.129442724",
      "y": "0.699450090"
    },
    {
      "turn": "-0.324919696",
      "x": "0.351549862",
      "y": "0.847830953"
    },
    {
      "turn": "0.000000000",
      "x": "0.616232451",
      "y": "0.845620609"
    },
    {
      "turn": "0.000000000",
      "x": "0.459411461",
      "y": "0.517774346"
    },
    {
      "turn": "0.000000000",
      "x": "0.546197369",
      "y": "0.221886609"
    },
    {
      "turn": "-0.324919696",
      "x": "0.863310944",
      "y": "0.733487348"
    },
    {
      "turn": "0.000000000",
      "x": "0.839999996",
      "y": "0.152169047"
    },
    {
      "turn": "0.000000000",
      "x": "0.750309572",
      "y": "0.437606168"
    }
  ],
  "radius": "0.160000000"
}

DISCUSSION

Discussion

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