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P86 · Packing and covering · Classic · Hard

Regular pentagons in a square · n = 21

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

Instancen = 21
ObjectiveMinimize container square side
Best known, unproven6.85368Source-precision interval: 6.85368–6.85369 (values inside match)exhibited site certificate 6.853686761Compiled by Erich Friedman; Found by Jake Loyd in June 2026. Source expression: s = 6.85368+.

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 = 21, 6.853686761
Current leader

6.853686761

container square side

Matches the best known
Answer sourceJake Loyd
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": "-0.158384440",
      "x": "0.121500823",
      "y": "0.899588209"
    },
    {
      "turn": "-0.158384440",
      "x": "0.439695407",
      "y": "0.899588209"
    },
    {
      "turn": "0.324517598",
      "x": "0.675098220",
      "y": "0.881930989"
    },
    {
      "turn": "0.324919696",
      "x": "0.899588209",
      "y": "0.830207584"
    },
    {
      "turn": "0.158384440",
      "x": "0.305136370",
      "y": "0.749119473"
    },
    {
      "turn": "0.154543984",
      "x": "0.530848004",
      "y": "0.717213197"
    },
    {
      "turn": "-0.000363744",
      "x": "0.749541742",
      "y": "0.694950730"
    },
    {
      "turn": "-0.158384440",
      "x": "0.118041140",
      "y": "0.675060618"
    },
    {
      "turn": "0.324517598",
      "x": "0.899535163",
      "y": "0.559763632"
    },
    {
      "turn": "-0.324131373",
      "x": "0.283297585",
      "y": "0.530721371"
    },
    {
      "turn": "0.229161301",
      "x": "0.505890019",
      "y": "0.494148954"
    },
    {
      "turn": "-0.001457407",
      "x": "0.716833897",
      "y": "0.469466618"
    },
    {
      "turn": "-0.000000000",
      "x": "0.100411791",
      "y": "0.439637491"
    },
    {
      "turn": "0.154937620",
      "x": "0.881703470",
      "y": "0.324587501"
    },
    {
      "turn": "-0.324919696",
      "x": "0.250935609",
      "y": "0.305154267"
    },
    {
      "turn": "-0.150450887",
      "x": "0.468780000",
      "y": "0.284660292"
    },
    {
      "turn": "-0.161834936",
      "x": "0.694267038",
      "y": "0.251307717"
    },
    {
      "turn": "0.000000000",
      "x": "0.100411791",
      "y": "0.121473922"
    },
    {
      "turn": "0.000000000",
      "x": "0.324939382",
      "y": "0.118041140"
    },
    {
      "turn": "0.154937620",
      "x": "0.559848591",
      "y": "0.100900382"
    },
    {
      "turn": "0.158384440",
      "x": "0.829069339",
      "y": "0.100411791"
    }
  ],
  "radius": "0.124115799"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 21,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "-0.158384440",
      "x": "0.121500823",
      "y": "0.899588209"
    },
    {
      "turn": "-0.158384440",
      "x": "0.439695407",
      "y": "0.899588209"
    },
    {
      "turn": "0.324517598",
      "x": "0.675098220",
      "y": "0.881930989"
    },
    {
      "turn": "0.324919696",
      "x": "0.899588209",
      "y": "0.830207584"
    },
    {
      "turn": "0.158384440",
      "x": "0.305136370",
      "y": "0.749119473"
    },
    {
      "turn": "0.154543984",
      "x": "0.530848004",
      "y": "0.717213197"
    },
    {
      "turn": "-0.000363744",
      "x": "0.749541742",
      "y": "0.694950730"
    },
    {
      "turn": "-0.158384440",
      "x": "0.118041140",
      "y": "0.675060618"
    },
    {
      "turn": "0.324517598",
      "x": "0.899535163",
      "y": "0.559763632"
    },
    {
      "turn": "-0.324131373",
      "x": "0.283297585",
      "y": "0.530721371"
    },
    {
      "turn": "0.229161301",
      "x": "0.505890019",
      "y": "0.494148954"
    },
    {
      "turn": "-0.001457407",
      "x": "0.716833897",
      "y": "0.469466618"
    },
    {
      "turn": "-0.000000000",
      "x": "0.100411791",
      "y": "0.439637491"
    },
    {
      "turn": "0.154937620",
      "x": "0.881703470",
      "y": "0.324587501"
    },
    {
      "turn": "-0.324919696",
      "x": "0.250935609",
      "y": "0.305154267"
    },
    {
      "turn": "-0.150450887",
      "x": "0.468780000",
      "y": "0.284660292"
    },
    {
      "turn": "-0.161834936",
      "x": "0.694267038",
      "y": "0.251307717"
    },
    {
      "turn": "0.000000000",
      "x": "0.100411791",
      "y": "0.121473922"
    },
    {
      "turn": "0.000000000",
      "x": "0.324939382",
      "y": "0.118041140"
    },
    {
      "turn": "0.154937620",
      "x": "0.559848591",
      "y": "0.100900382"
    },
    {
      "turn": "0.158384440",
      "x": "0.829069339",
      "y": "0.100411791"
    }
  ],
  "radius": "0.124115799"
}

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

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