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P86 · Packing and covering · Classic · Hard

Regular pentagons in a square · n = 23

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

Instancen = 23
ObjectiveMinimize container square side
Best known, unproven7.20624Source-precision interval: 7.20624–7.20625 (values inside match)exhibited site certificate 7.206241778Compiled by Erich Friedman; Found by Jake Loyd in June 2026. Source expression: s = 7.20624+.

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 = 23, 7.206241778
Current leader

7.206241778

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": "-0.093541270",
      "x": "0.106710550",
      "y": "0.896446567"
    },
    {
      "turn": "-0.158384440",
      "x": "0.425107935",
      "y": "0.904500711"
    },
    {
      "turn": "-0.158384440",
      "x": "0.649640231",
      "y": "0.904500711"
    },
    {
      "turn": "0.246617058",
      "x": "0.895490314",
      "y": "0.851271108"
    },
    {
      "turn": "-0.324919696",
      "x": "0.262013850",
      "y": "0.766183296"
    },
    {
      "turn": "0.074205882",
      "x": "0.732904210",
      "y": "0.721249244"
    },
    {
      "turn": "-0.269352997",
      "x": "0.537374083",
      "y": "0.725381708"
    },
    {
      "turn": "0.000000000",
      "x": "0.095499289",
      "y": "0.670424736"
    },
    {
      "turn": "-0.324919696",
      "x": "0.904500711",
      "y": "0.617333496"
    },
    {
      "turn": "0.051094965",
      "x": "0.388010711",
      "y": "0.605866205"
    },
    {
      "turn": "0.313423695",
      "x": "0.557416135",
      "y": "0.501736356"
    },
    {
      "turn": "-0.010433481",
      "x": "0.748517163",
      "y": "0.504652666"
    },
    {
      "turn": "-0.293243776",
      "x": "0.204936110",
      "y": "0.489554937"
    },
    {
      "turn": "0.313423695",
      "x": "0.903073818",
      "y": "0.392364595"
    },
    {
      "turn": "-0.040238689",
      "x": "0.397254559",
      "y": "0.390571519"
    },
    {
      "turn": "0.028920368",
      "x": "0.099349559",
      "y": "0.317026483"
    },
    {
      "turn": "-0.040238689",
      "x": "0.741228890",
      "y": "0.283621382"
    },
    {
      "turn": "0.281007025",
      "x": "0.548828745",
      "y": "0.273919464"
    },
    {
      "turn": "-0.293243776",
      "x": "0.268981928",
      "y": "0.227719294"
    },
    {
      "turn": "0.066656204",
      "x": "0.882982696",
      "y": "0.124928945"
    },
    {
      "turn": "0.150476064",
      "x": "0.111689261",
      "y": "0.096559718"
    },
    {
      "turn": "0.158384440",
      "x": "0.436562597",
      "y": "0.095499289"
    },
    {
      "turn": "0.158384440",
      "x": "0.661094893",
      "y": "0.095499289"
    }
  ],
  "radius": "0.118043612"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 23,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "-0.093541270",
      "x": "0.106710550",
      "y": "0.896446567"
    },
    {
      "turn": "-0.158384440",
      "x": "0.425107935",
      "y": "0.904500711"
    },
    {
      "turn": "-0.158384440",
      "x": "0.649640231",
      "y": "0.904500711"
    },
    {
      "turn": "0.246617058",
      "x": "0.895490314",
      "y": "0.851271108"
    },
    {
      "turn": "-0.324919696",
      "x": "0.262013850",
      "y": "0.766183296"
    },
    {
      "turn": "0.074205882",
      "x": "0.732904210",
      "y": "0.721249244"
    },
    {
      "turn": "-0.269352997",
      "x": "0.537374083",
      "y": "0.725381708"
    },
    {
      "turn": "0.000000000",
      "x": "0.095499289",
      "y": "0.670424736"
    },
    {
      "turn": "-0.324919696",
      "x": "0.904500711",
      "y": "0.617333496"
    },
    {
      "turn": "0.051094965",
      "x": "0.388010711",
      "y": "0.605866205"
    },
    {
      "turn": "0.313423695",
      "x": "0.557416135",
      "y": "0.501736356"
    },
    {
      "turn": "-0.010433481",
      "x": "0.748517163",
      "y": "0.504652666"
    },
    {
      "turn": "-0.293243776",
      "x": "0.204936110",
      "y": "0.489554937"
    },
    {
      "turn": "0.313423695",
      "x": "0.903073818",
      "y": "0.392364595"
    },
    {
      "turn": "-0.040238689",
      "x": "0.397254559",
      "y": "0.390571519"
    },
    {
      "turn": "0.028920368",
      "x": "0.099349559",
      "y": "0.317026483"
    },
    {
      "turn": "-0.040238689",
      "x": "0.741228890",
      "y": "0.283621382"
    },
    {
      "turn": "0.281007025",
      "x": "0.548828745",
      "y": "0.273919464"
    },
    {
      "turn": "-0.293243776",
      "x": "0.268981928",
      "y": "0.227719294"
    },
    {
      "turn": "0.066656204",
      "x": "0.882982696",
      "y": "0.124928945"
    },
    {
      "turn": "0.150476064",
      "x": "0.111689261",
      "y": "0.096559718"
    },
    {
      "turn": "0.158384440",
      "x": "0.436562597",
      "y": "0.095499289"
    },
    {
      "turn": "0.158384440",
      "x": "0.661094893",
      "y": "0.095499289"
    }
  ],
  "radius": "0.118043612"
}

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

DISCUSSION

Discussion

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