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P86 · Packing and covering · Classic · Hard

Regular pentagons in a square · n = 24

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

Instancen = 24
ObjectiveMinimize container square side
Best known, unproven7.33690Source-precision interval: 7.3369–7.33691 (values inside match)exhibited site certificate 7.336904101Compiled by Erich Friedman; Found by Jake Loyd in June 2026. Source expression: s = 7.33690+.

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 = 24, 7.336904101
Current leader

7.336904101

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": "0.296181314",
      "y": "0.906201450"
    },
    {
      "turn": "-0.158384440",
      "x": "0.524453858",
      "y": "0.906201450"
    },
    {
      "turn": "-0.000000000",
      "x": "0.728519217",
      "y": "0.889733191"
    },
    {
      "turn": "-0.000000000",
      "x": "0.093798550",
      "y": "0.837580703"
    },
    {
      "turn": "0.324919696",
      "x": "0.906201450",
      "y": "0.815132555"
    },
    {
      "turn": "0.158384440",
      "x": "0.410317586",
      "y": "0.757243537"
    },
    {
      "turn": "-0.324919696",
      "x": "0.217864562",
      "y": "0.689183669"
    },
    {
      "turn": "0.000000000",
      "x": "0.770447318",
      "y": "0.682822840"
    },
    {
      "turn": "0.324919696",
      "x": "0.582850218",
      "y": "0.647537607"
    },
    {
      "turn": "-0.158384440",
      "x": "0.393591577",
      "y": "0.569646437"
    },
    {
      "turn": "-0.324919696",
      "x": "0.906201450",
      "y": "0.550513124"
    },
    {
      "turn": "-0.000000000",
      "x": "0.093798550",
      "y": "0.540786635"
    },
    {
      "turn": "-0.000000000",
      "x": "0.731881517",
      "y": "0.471284678"
    },
    {
      "turn": "0.158384440",
      "x": "0.527816159",
      "y": "0.435283536"
    },
    {
      "turn": "0.158384440",
      "x": "0.265923679",
      "y": "0.430519826"
    },
    {
      "turn": "0.222391217",
      "x": "0.894986567",
      "y": "0.328544586"
    },
    {
      "turn": "-0.000000000",
      "x": "0.093798550",
      "y": "0.320253018"
    },
    {
      "turn": "-0.095619111",
      "x": "0.719689372",
      "y": "0.260538577"
    },
    {
      "turn": "-0.158384440",
      "x": "0.504497605",
      "y": "0.247686436"
    },
    {
      "turn": "-0.158384440",
      "x": "0.283239431",
      "y": "0.242922726"
    },
    {
      "turn": "0.159906789",
      "x": "0.149819144",
      "y": "0.107223119"
    },
    {
      "turn": "-0.195247848",
      "x": "0.869370224",
      "y": "0.115645879"
    },
    {
      "turn": "0.158384440",
      "x": "0.397146860",
      "y": "0.093798550"
    },
    {
      "turn": "0.158384440",
      "x": "0.617680478",
      "y": "0.093798550"
    }
  ],
  "radius": "0.115941383"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 24,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "-0.158384440",
      "x": "0.296181314",
      "y": "0.906201450"
    },
    {
      "turn": "-0.158384440",
      "x": "0.524453858",
      "y": "0.906201450"
    },
    {
      "turn": "-0.000000000",
      "x": "0.728519217",
      "y": "0.889733191"
    },
    {
      "turn": "-0.000000000",
      "x": "0.093798550",
      "y": "0.837580703"
    },
    {
      "turn": "0.324919696",
      "x": "0.906201450",
      "y": "0.815132555"
    },
    {
      "turn": "0.158384440",
      "x": "0.410317586",
      "y": "0.757243537"
    },
    {
      "turn": "-0.324919696",
      "x": "0.217864562",
      "y": "0.689183669"
    },
    {
      "turn": "0.000000000",
      "x": "0.770447318",
      "y": "0.682822840"
    },
    {
      "turn": "0.324919696",
      "x": "0.582850218",
      "y": "0.647537607"
    },
    {
      "turn": "-0.158384440",
      "x": "0.393591577",
      "y": "0.569646437"
    },
    {
      "turn": "-0.324919696",
      "x": "0.906201450",
      "y": "0.550513124"
    },
    {
      "turn": "-0.000000000",
      "x": "0.093798550",
      "y": "0.540786635"
    },
    {
      "turn": "-0.000000000",
      "x": "0.731881517",
      "y": "0.471284678"
    },
    {
      "turn": "0.158384440",
      "x": "0.527816159",
      "y": "0.435283536"
    },
    {
      "turn": "0.158384440",
      "x": "0.265923679",
      "y": "0.430519826"
    },
    {
      "turn": "0.222391217",
      "x": "0.894986567",
      "y": "0.328544586"
    },
    {
      "turn": "-0.000000000",
      "x": "0.093798550",
      "y": "0.320253018"
    },
    {
      "turn": "-0.095619111",
      "x": "0.719689372",
      "y": "0.260538577"
    },
    {
      "turn": "-0.158384440",
      "x": "0.504497605",
      "y": "0.247686436"
    },
    {
      "turn": "-0.158384440",
      "x": "0.283239431",
      "y": "0.242922726"
    },
    {
      "turn": "0.159906789",
      "x": "0.149819144",
      "y": "0.107223119"
    },
    {
      "turn": "-0.195247848",
      "x": "0.869370224",
      "y": "0.115645879"
    },
    {
      "turn": "0.158384440",
      "x": "0.397146860",
      "y": "0.093798550"
    },
    {
      "turn": "0.158384440",
      "x": "0.617680478",
      "y": "0.093798550"
    }
  ],
  "radius": "0.115941383"
}

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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