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P86 · Packing and covering · Classic · Hard

Regular pentagons in a square · n = 27

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

Instancen = 27
ObjectiveMinimize container square side
Best known, unproven7.67948Source-precision interval: 7.67948–7.67949 (values inside match)exhibited site certificate 7.679480071Compiled by Erich Friedman; Found by Jonathan Viquerat in August 2026. Source expression: s = 7.67948+.

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 = 27, 7.679480071
Current leader

7.679480071

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": "-0.161566150",
      "x": "0.180335566",
      "y": "0.909312781"
    },
    {
      "turn": "-0.232709085",
      "x": "0.391890733",
      "y": "0.902015387"
    },
    {
      "turn": "-0.158384440",
      "x": "0.592646436",
      "y": "0.910385734"
    },
    {
      "turn": "-0.158384440",
      "x": "0.824383341",
      "y": "0.910385734"
    },
    {
      "turn": "0.158384440",
      "x": "0.708514888",
      "y": "0.773030445"
    },
    {
      "turn": "0.085728526",
      "x": "0.288017813",
      "y": "0.753026023"
    },
    {
      "turn": "0.085728526",
      "x": "0.491554775",
      "y": "0.751948159"
    },
    {
      "turn": "0.000000000",
      "x": "0.089614266",
      "y": "0.733880068"
    },
    {
      "turn": "-0.324919696",
      "x": "0.910385734",
      "y": "0.731883657"
    },
    {
      "turn": "0.324919696",
      "x": "0.609682533",
      "y": "0.592986296"
    },
    {
      "turn": "0.000000000",
      "x": "0.788911064",
      "y": "0.594157407"
    },
    {
      "turn": "-0.324919696",
      "x": "0.201178057",
      "y": "0.582512663"
    },
    {
      "turn": "-0.232709085",
      "x": "0.404321585",
      "y": "0.583439018"
    },
    {
      "turn": "0.324919696",
      "x": "0.910385734",
      "y": "0.456431157"
    },
    {
      "turn": "0.085728526",
      "x": "0.503162116",
      "y": "0.432937605"
    },
    {
      "turn": "-0.000161416",
      "x": "0.089635280",
      "y": "0.430971530"
    },
    {
      "turn": "0.000000000",
      "x": "0.293187217",
      "y": "0.423956222"
    },
    {
      "turn": "-0.130131978",
      "x": "0.695950321",
      "y": "0.417545396"
    },
    {
      "turn": "0.258140571",
      "x": "0.821000546",
      "y": "0.277943125"
    },
    {
      "turn": "-0.324919696",
      "x": "0.192918618",
      "y": "0.268083435"
    },
    {
      "turn": "0.291338307",
      "x": "0.401029413",
      "y": "0.267019191"
    },
    {
      "turn": "-0.135867920",
      "x": "0.590925644",
      "y": "0.246843936"
    },
    {
      "turn": "0.000000000",
      "x": "0.089614266",
      "y": "0.105347887"
    },
    {
      "turn": "-0.030677417",
      "x": "0.290283375",
      "y": "0.107247964"
    },
    {
      "turn": "0.258140571",
      "x": "0.708434549",
      "y": "0.100349195"
    },
    {
      "turn": "-0.061611466",
      "x": "0.890068451",
      "y": "0.108753038"
    },
    {
      "turn": "0.181058729",
      "x": "0.499551673",
      "y": "0.092396105"
    }
  ],
  "radius": "0.110769323"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 27,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "-0.161566150",
      "x": "0.180335566",
      "y": "0.909312781"
    },
    {
      "turn": "-0.232709085",
      "x": "0.391890733",
      "y": "0.902015387"
    },
    {
      "turn": "-0.158384440",
      "x": "0.592646436",
      "y": "0.910385734"
    },
    {
      "turn": "-0.158384440",
      "x": "0.824383341",
      "y": "0.910385734"
    },
    {
      "turn": "0.158384440",
      "x": "0.708514888",
      "y": "0.773030445"
    },
    {
      "turn": "0.085728526",
      "x": "0.288017813",
      "y": "0.753026023"
    },
    {
      "turn": "0.085728526",
      "x": "0.491554775",
      "y": "0.751948159"
    },
    {
      "turn": "0.000000000",
      "x": "0.089614266",
      "y": "0.733880068"
    },
    {
      "turn": "-0.324919696",
      "x": "0.910385734",
      "y": "0.731883657"
    },
    {
      "turn": "0.324919696",
      "x": "0.609682533",
      "y": "0.592986296"
    },
    {
      "turn": "0.000000000",
      "x": "0.788911064",
      "y": "0.594157407"
    },
    {
      "turn": "-0.324919696",
      "x": "0.201178057",
      "y": "0.582512663"
    },
    {
      "turn": "-0.232709085",
      "x": "0.404321585",
      "y": "0.583439018"
    },
    {
      "turn": "0.324919696",
      "x": "0.910385734",
      "y": "0.456431157"
    },
    {
      "turn": "0.085728526",
      "x": "0.503162116",
      "y": "0.432937605"
    },
    {
      "turn": "-0.000161416",
      "x": "0.089635280",
      "y": "0.430971530"
    },
    {
      "turn": "0.000000000",
      "x": "0.293187217",
      "y": "0.423956222"
    },
    {
      "turn": "-0.130131978",
      "x": "0.695950321",
      "y": "0.417545396"
    },
    {
      "turn": "0.258140571",
      "x": "0.821000546",
      "y": "0.277943125"
    },
    {
      "turn": "-0.324919696",
      "x": "0.192918618",
      "y": "0.268083435"
    },
    {
      "turn": "0.291338307",
      "x": "0.401029413",
      "y": "0.267019191"
    },
    {
      "turn": "-0.135867920",
      "x": "0.590925644",
      "y": "0.246843936"
    },
    {
      "turn": "0.000000000",
      "x": "0.089614266",
      "y": "0.105347887"
    },
    {
      "turn": "-0.030677417",
      "x": "0.290283375",
      "y": "0.107247964"
    },
    {
      "turn": "0.258140571",
      "x": "0.708434549",
      "y": "0.100349195"
    },
    {
      "turn": "-0.061611466",
      "x": "0.890068451",
      "y": "0.108753038"
    },
    {
      "turn": "0.181058729",
      "x": "0.499551673",
      "y": "0.092396105"
    }
  ],
  "radius": "0.110769323"
}

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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