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P86 · Packing and covering · Classic · Hard

Regular pentagons in a square · n = 13

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

Instancen = 13
ObjectiveMinimize container square side
Best known, unproven5.52102Source-precision interval: 5.52102–5.52103 (values inside match)exhibited site certificate 5.521022435Compiled by Erich Friedman; Found by Jake Loyd in June 2026. Source expression: s = 5.52102+.

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 = 13, 5.521022435
Current leader

5.521022435

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": "-0.217289213",
      "x": "0.332029526",
      "y": "0.865875733"
    },
    {
      "turn": "-0.158384440",
      "x": "0.613504221",
      "y": "0.875350812"
    },
    {
      "turn": "0.324919696",
      "x": "0.875350812",
      "y": "0.774387484"
    },
    {
      "turn": "0.100532718",
      "x": "0.140181665",
      "y": "0.703551931"
    },
    {
      "turn": "0.107901299",
      "x": "0.409122147",
      "y": "0.621577791"
    },
    {
      "turn": "-0.000000000",
      "x": "0.684299803",
      "y": "0.613214908"
    },
    {
      "turn": "0.324919696",
      "x": "0.875350812",
      "y": "0.452042333"
    },
    {
      "turn": "-0.000000000",
      "x": "0.124649188",
      "y": "0.427892401"
    },
    {
      "turn": "-0.209667607",
      "x": "0.390707333",
      "y": "0.372880576"
    },
    {
      "turn": "0.000000000",
      "x": "0.656455404",
      "y": "0.329194284"
    },
    {
      "turn": "0.271188449",
      "x": "0.865092900",
      "y": "0.156410082"
    },
    {
      "turn": "-0.181230951",
      "x": "0.151681002",
      "y": "0.157420616"
    },
    {
      "turn": "0.158384440",
      "x": "0.476623334",
      "y": "0.124649188"
    }
  ],
  "radius": "0.154074869"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 13,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "-0.217289213",
      "x": "0.332029526",
      "y": "0.865875733"
    },
    {
      "turn": "-0.158384440",
      "x": "0.613504221",
      "y": "0.875350812"
    },
    {
      "turn": "0.324919696",
      "x": "0.875350812",
      "y": "0.774387484"
    },
    {
      "turn": "0.100532718",
      "x": "0.140181665",
      "y": "0.703551931"
    },
    {
      "turn": "0.107901299",
      "x": "0.409122147",
      "y": "0.621577791"
    },
    {
      "turn": "-0.000000000",
      "x": "0.684299803",
      "y": "0.613214908"
    },
    {
      "turn": "0.324919696",
      "x": "0.875350812",
      "y": "0.452042333"
    },
    {
      "turn": "-0.000000000",
      "x": "0.124649188",
      "y": "0.427892401"
    },
    {
      "turn": "-0.209667607",
      "x": "0.390707333",
      "y": "0.372880576"
    },
    {
      "turn": "0.000000000",
      "x": "0.656455404",
      "y": "0.329194284"
    },
    {
      "turn": "0.271188449",
      "x": "0.865092900",
      "y": "0.156410082"
    },
    {
      "turn": "-0.181230951",
      "x": "0.151681002",
      "y": "0.157420616"
    },
    {
      "turn": "0.158384440",
      "x": "0.476623334",
      "y": "0.124649188"
    }
  ],
  "radius": "0.154074869"
}

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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