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P86 · Packing and covering · Classic · Hard

Regular pentagons in a square · n = 14

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

Instancen = 14
ObjectiveMinimize container square side
Best known, unproven5.69659Source-precision interval: 5.69659–5.6966 (values inside match)exhibited site certificate 5.696593615Compiled by Erich Friedman; Found by Jake Loyd in June 2026. Source expression: s = 5.69659+.

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 = 14, 5.696593615
Current leader

5.696593615

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": "0.288143357",
      "x": "0.381330754",
      "y": "0.855203182"
    },
    {
      "turn": "-0.158384440",
      "x": "0.642937682",
      "y": "0.879192547"
    },
    {
      "turn": "-0.000000000",
      "x": "0.120807453",
      "y": "0.805101520"
    },
    {
      "turn": "0.324919696",
      "x": "0.879192547",
      "y": "0.757224650"
    },
    {
      "turn": "-0.033627973",
      "x": "0.677629743",
      "y": "0.614024806"
    },
    {
      "turn": "-0.033627973",
      "x": "0.406778037",
      "y": "0.612444862"
    },
    {
      "turn": "-0.033627973",
      "x": "0.126431038",
      "y": "0.528283213"
    },
    {
      "turn": "0.288143357",
      "x": "0.873568962",
      "y": "0.471716787"
    },
    {
      "turn": "0.288143357",
      "x": "0.593221963",
      "y": "0.387555138"
    },
    {
      "turn": "0.288143357",
      "x": "0.322370257",
      "y": "0.385975194"
    },
    {
      "turn": "-0.000000000",
      "x": "0.120807453",
      "y": "0.242775350"
    },
    {
      "turn": "-0.324919696",
      "x": "0.879192547",
      "y": "0.194898480"
    },
    {
      "turn": "-0.033627973",
      "x": "0.618669246",
      "y": "0.144796818"
    },
    {
      "turn": "0.158384440",
      "x": "0.357062318",
      "y": "0.120807453"
    }
  ],
  "radius": "0.149326223"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 14,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.288143357",
      "x": "0.381330754",
      "y": "0.855203182"
    },
    {
      "turn": "-0.158384440",
      "x": "0.642937682",
      "y": "0.879192547"
    },
    {
      "turn": "-0.000000000",
      "x": "0.120807453",
      "y": "0.805101520"
    },
    {
      "turn": "0.324919696",
      "x": "0.879192547",
      "y": "0.757224650"
    },
    {
      "turn": "-0.033627973",
      "x": "0.677629743",
      "y": "0.614024806"
    },
    {
      "turn": "-0.033627973",
      "x": "0.406778037",
      "y": "0.612444862"
    },
    {
      "turn": "-0.033627973",
      "x": "0.126431038",
      "y": "0.528283213"
    },
    {
      "turn": "0.288143357",
      "x": "0.873568962",
      "y": "0.471716787"
    },
    {
      "turn": "0.288143357",
      "x": "0.593221963",
      "y": "0.387555138"
    },
    {
      "turn": "0.288143357",
      "x": "0.322370257",
      "y": "0.385975194"
    },
    {
      "turn": "-0.000000000",
      "x": "0.120807453",
      "y": "0.242775350"
    },
    {
      "turn": "-0.324919696",
      "x": "0.879192547",
      "y": "0.194898480"
    },
    {
      "turn": "-0.033627973",
      "x": "0.618669246",
      "y": "0.144796818"
    },
    {
      "turn": "0.158384440",
      "x": "0.357062318",
      "y": "0.120807453"
    }
  ],
  "radius": "0.149326223"
}

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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