MinMax Arena
中/EN
P86 · Packing and covering · Classic · Hard

Regular pentagons in a square · n = 26

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

Instancen = 26
ObjectiveMinimize container square side
Best known, unproven7.58176Source-precision interval: 7.58176–7.58177 (values inside match)exhibited site certificate 7.581760052Compiled by Erich Friedman; Found by Jonathan Viquerat in August 2026. Source expression: s = 7.58176+.

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 = 26, 7.581760052
Current leader

7.581760052

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": "-0.168860793",
      "x": "0.182838755",
      "y": "0.907903957"
    },
    {
      "turn": "-0.170482844",
      "x": "0.394948832",
      "y": "0.907702347"
    },
    {
      "turn": "-0.158384440",
      "x": "0.606307896",
      "y": "0.909230712"
    },
    {
      "turn": "-0.155334113",
      "x": "0.819184412",
      "y": "0.908839660"
    },
    {
      "turn": "0.146331100",
      "x": "0.288895981",
      "y": "0.760272943"
    },
    {
      "turn": "0.158384440",
      "x": "0.711749751",
      "y": "0.761444682"
    },
    {
      "turn": "0.000000000",
      "x": "0.500768207",
      "y": "0.741817276"
    },
    {
      "turn": "-0.000000000",
      "x": "0.090769288",
      "y": "0.730808173"
    },
    {
      "turn": "-0.324919696",
      "x": "0.909230712",
      "y": "0.730186102"
    },
    {
      "turn": "0.324919696",
      "x": "0.612091119",
      "y": "0.586188246"
    },
    {
      "turn": "-0.324919696",
      "x": "0.205942482",
      "y": "0.580478601"
    },
    {
      "turn": "0.324919696",
      "x": "0.408908781",
      "y": "0.580783257"
    },
    {
      "turn": "-0.000000000",
      "x": "0.887802989",
      "y": "0.532342857"
    },
    {
      "turn": "0.000000000",
      "x": "0.710176696",
      "y": "0.427177226"
    },
    {
      "turn": "0.007024949",
      "x": "0.094360967",
      "y": "0.423342484"
    },
    {
      "turn": "-0.000000000",
      "x": "0.300983174",
      "y": "0.420478237"
    },
    {
      "turn": "0.000000000",
      "x": "0.503949473",
      "y": "0.420782894"
    },
    {
      "turn": "0.324919696",
      "x": "0.875256673",
      "y": "0.345538447"
    },
    {
      "turn": "-0.324919696",
      "x": "0.616435428",
      "y": "0.266754654"
    },
    {
      "turn": "0.324919696",
      "x": "0.202668372",
      "y": "0.261541697"
    },
    {
      "turn": "-0.324919696",
      "x": "0.409266926",
      "y": "0.260666162"
    },
    {
      "turn": "0.324919696",
      "x": "0.909230712",
      "y": "0.143165885"
    },
    {
      "turn": "-0.000000000",
      "x": "0.090769288",
      "y": "0.106705698"
    },
    {
      "turn": "-0.000000000",
      "x": "0.711326566",
      "y": "0.106705698"
    },
    {
      "turn": "0.126288263",
      "x": "0.511882884",
      "y": "0.094735935"
    },
    {
      "turn": "0.158384439",
      "x": "0.303584830",
      "y": "0.090769289"
    }
  ],
  "radius": "0.112197010"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 26,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "-0.168860793",
      "x": "0.182838755",
      "y": "0.907903957"
    },
    {
      "turn": "-0.170482844",
      "x": "0.394948832",
      "y": "0.907702347"
    },
    {
      "turn": "-0.158384440",
      "x": "0.606307896",
      "y": "0.909230712"
    },
    {
      "turn": "-0.155334113",
      "x": "0.819184412",
      "y": "0.908839660"
    },
    {
      "turn": "0.146331100",
      "x": "0.288895981",
      "y": "0.760272943"
    },
    {
      "turn": "0.158384440",
      "x": "0.711749751",
      "y": "0.761444682"
    },
    {
      "turn": "0.000000000",
      "x": "0.500768207",
      "y": "0.741817276"
    },
    {
      "turn": "-0.000000000",
      "x": "0.090769288",
      "y": "0.730808173"
    },
    {
      "turn": "-0.324919696",
      "x": "0.909230712",
      "y": "0.730186102"
    },
    {
      "turn": "0.324919696",
      "x": "0.612091119",
      "y": "0.586188246"
    },
    {
      "turn": "-0.324919696",
      "x": "0.205942482",
      "y": "0.580478601"
    },
    {
      "turn": "0.324919696",
      "x": "0.408908781",
      "y": "0.580783257"
    },
    {
      "turn": "-0.000000000",
      "x": "0.887802989",
      "y": "0.532342857"
    },
    {
      "turn": "0.000000000",
      "x": "0.710176696",
      "y": "0.427177226"
    },
    {
      "turn": "0.007024949",
      "x": "0.094360967",
      "y": "0.423342484"
    },
    {
      "turn": "-0.000000000",
      "x": "0.300983174",
      "y": "0.420478237"
    },
    {
      "turn": "0.000000000",
      "x": "0.503949473",
      "y": "0.420782894"
    },
    {
      "turn": "0.324919696",
      "x": "0.875256673",
      "y": "0.345538447"
    },
    {
      "turn": "-0.324919696",
      "x": "0.616435428",
      "y": "0.266754654"
    },
    {
      "turn": "0.324919696",
      "x": "0.202668372",
      "y": "0.261541697"
    },
    {
      "turn": "-0.324919696",
      "x": "0.409266926",
      "y": "0.260666162"
    },
    {
      "turn": "0.324919696",
      "x": "0.909230712",
      "y": "0.143165885"
    },
    {
      "turn": "-0.000000000",
      "x": "0.090769288",
      "y": "0.106705698"
    },
    {
      "turn": "-0.000000000",
      "x": "0.711326566",
      "y": "0.106705698"
    },
    {
      "turn": "0.126288263",
      "x": "0.511882884",
      "y": "0.094735935"
    },
    {
      "turn": "0.158384439",
      "x": "0.303584830",
      "y": "0.090769289"
    }
  ],
  "radius": "0.112197010"
}

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

Talk strategy, share methods, ask why you are stuck. Every signed-in user can post. Posts carry your public byline, the same name your records use; the #number after it is the account's signup ordinal, so a name cannot be worn by someone else. New posts appear after an automated review.

No threads yet. Be the first to talk about this problem?