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P86 · Packing and covering · Classic · Hard

Regular pentagons in a square · n = 28

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

Instancen = 28
ObjectiveMinimize container square side
Best known, unproven7.82010Source-precision interval: 7.8201–7.82011 (values inside match)exhibited site certificate 7.820100053Compiled by Erich Friedman; Found by Jonathan Viquerat in August 2026. Source expression: s = 7.82010+.

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 = 28, 7.820100053
Current leader

7.820100053

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": "-0.249016834",
      "x": "0.107709980",
      "y": "0.902259384"
    },
    {
      "turn": "-0.225071090",
      "x": "0.306008063",
      "y": "0.904523843"
    },
    {
      "turn": "0.324919696",
      "x": "0.911997166",
      "y": "0.896546464"
    },
    {
      "turn": "-0.209281827",
      "x": "0.505871809",
      "y": "0.906138830"
    },
    {
      "turn": "-0.158384433",
      "x": "0.705497880",
      "y": "0.911997162"
    },
    {
      "turn": "-0.062363395",
      "x": "0.203038771",
      "y": "0.737734802"
    },
    {
      "turn": "-0.062363504",
      "x": "0.403548068",
      "y": "0.739800326"
    },
    {
      "turn": "-0.062363504",
      "x": "0.601825449",
      "y": "0.739484056"
    },
    {
      "turn": "-0.056390391",
      "x": "0.799921639",
      "y": "0.740086505"
    },
    {
      "turn": "0.263697637",
      "x": "0.905366960",
      "y": "0.594596579"
    },
    {
      "turn": "0.257341657",
      "x": "0.107934647",
      "y": "0.587941666"
    },
    {
      "turn": "0.257341645",
      "x": "0.306617523",
      "y": "0.590864209"
    },
    {
      "turn": "0.257341645",
      "x": "0.504895006",
      "y": "0.590547891"
    },
    {
      "turn": "0.257341645",
      "x": "0.703172324",
      "y": "0.590231504"
    },
    {
      "turn": "-0.062363504",
      "x": "0.693382482",
      "y": "0.409135775"
    },
    {
      "turn": "-0.062363342",
      "x": "0.892065349",
      "y": "0.412058315"
    },
    {
      "turn": "-0.062363504",
      "x": "0.296827697",
      "y": "0.409768483"
    },
    {
      "turn": "-0.062363504",
      "x": "0.495105019",
      "y": "0.409452134"
    },
    {
      "turn": "-0.056390543",
      "x": "0.094633046",
      "y": "0.405403406"
    },
    {
      "turn": "0.257341645",
      "x": "0.398174566",
      "y": "0.260515974"
    },
    {
      "turn": "0.257341817",
      "x": "0.796961246",
      "y": "0.262265180"
    },
    {
      "turn": "0.263697637",
      "x": "0.200078369",
      "y": "0.259913502"
    },
    {
      "turn": "0.257341645",
      "x": "0.596451925",
      "y": "0.260199664"
    },
    {
      "turn": "0.000000000",
      "x": "0.088002834",
      "y": "0.103453536"
    },
    {
      "turn": "0.070221449",
      "x": "0.892290026",
      "y": "0.097740586"
    },
    {
      "turn": "0.108275095",
      "x": "0.494128161",
      "y": "0.093861179"
    },
    {
      "turn": "0.093044502",
      "x": "0.693991932",
      "y": "0.095476133"
    },
    {
      "turn": "0.158384451",
      "x": "0.294502101",
      "y": "0.088002835"
    }
  ],
  "radius": "0.108777484"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 28,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "-0.249016834",
      "x": "0.107709980",
      "y": "0.902259384"
    },
    {
      "turn": "-0.225071090",
      "x": "0.306008063",
      "y": "0.904523843"
    },
    {
      "turn": "0.324919696",
      "x": "0.911997166",
      "y": "0.896546464"
    },
    {
      "turn": "-0.209281827",
      "x": "0.505871809",
      "y": "0.906138830"
    },
    {
      "turn": "-0.158384433",
      "x": "0.705497880",
      "y": "0.911997162"
    },
    {
      "turn": "-0.062363395",
      "x": "0.203038771",
      "y": "0.737734802"
    },
    {
      "turn": "-0.062363504",
      "x": "0.403548068",
      "y": "0.739800326"
    },
    {
      "turn": "-0.062363504",
      "x": "0.601825449",
      "y": "0.739484056"
    },
    {
      "turn": "-0.056390391",
      "x": "0.799921639",
      "y": "0.740086505"
    },
    {
      "turn": "0.263697637",
      "x": "0.905366960",
      "y": "0.594596579"
    },
    {
      "turn": "0.257341657",
      "x": "0.107934647",
      "y": "0.587941666"
    },
    {
      "turn": "0.257341645",
      "x": "0.306617523",
      "y": "0.590864209"
    },
    {
      "turn": "0.257341645",
      "x": "0.504895006",
      "y": "0.590547891"
    },
    {
      "turn": "0.257341645",
      "x": "0.703172324",
      "y": "0.590231504"
    },
    {
      "turn": "-0.062363504",
      "x": "0.693382482",
      "y": "0.409135775"
    },
    {
      "turn": "-0.062363342",
      "x": "0.892065349",
      "y": "0.412058315"
    },
    {
      "turn": "-0.062363504",
      "x": "0.296827697",
      "y": "0.409768483"
    },
    {
      "turn": "-0.062363504",
      "x": "0.495105019",
      "y": "0.409452134"
    },
    {
      "turn": "-0.056390543",
      "x": "0.094633046",
      "y": "0.405403406"
    },
    {
      "turn": "0.257341645",
      "x": "0.398174566",
      "y": "0.260515974"
    },
    {
      "turn": "0.257341817",
      "x": "0.796961246",
      "y": "0.262265180"
    },
    {
      "turn": "0.263697637",
      "x": "0.200078369",
      "y": "0.259913502"
    },
    {
      "turn": "0.257341645",
      "x": "0.596451925",
      "y": "0.260199664"
    },
    {
      "turn": "0.000000000",
      "x": "0.088002834",
      "y": "0.103453536"
    },
    {
      "turn": "0.070221449",
      "x": "0.892290026",
      "y": "0.097740586"
    },
    {
      "turn": "0.108275095",
      "x": "0.494128161",
      "y": "0.093861179"
    },
    {
      "turn": "0.093044502",
      "x": "0.693991932",
      "y": "0.095476133"
    },
    {
      "turn": "0.158384451",
      "x": "0.294502101",
      "y": "0.088002835"
    }
  ],
  "radius": "0.108777484"
}

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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