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

Regular pentagons in a square · n = 25

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

Instancen = 25
ObjectiveMinimize container square side
Best known, unproven7.45360Source-precision interval: 7.4536–7.45361 (values inside match)exhibited site certificate 7.453600088Compiled by Erich Friedman; Found by Jake Loyd in June 2026. Source expression: s = 7.45360+.

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 = 25, 7.453600088
Current leader

7.453600088

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": "0.304220615",
      "y": "0.907669990"
    },
    {
      "turn": "-0.146427724",
      "x": "0.519229945",
      "y": "0.906127624"
    },
    {
      "turn": "-0.158384440",
      "x": "0.828487975",
      "y": "0.907669990"
    },
    {
      "turn": "-0.000084734",
      "x": "0.094264873",
      "y": "0.851662728"
    },
    {
      "turn": "0.170385498",
      "x": "0.672532878",
      "y": "0.790819194"
    },
    {
      "turn": "0.000000000",
      "x": "0.427698515",
      "y": "0.724238889"
    },
    {
      "turn": "0.324919696",
      "x": "0.907669990",
      "y": "0.716909566"
    },
    {
      "turn": "-0.324919696",
      "x": "0.243038496",
      "y": "0.716341580"
    },
    {
      "turn": "-0.324919696",
      "x": "0.573616877",
      "y": "0.610915894"
    },
    {
      "turn": "0.000000000",
      "x": "0.092330010",
      "y": "0.609611626"
    },
    {
      "turn": "-0.000000000",
      "x": "0.758276896",
      "y": "0.608369130"
    },
    {
      "turn": "0.000000000",
      "x": "0.425570166",
      "y": "0.500522322"
    },
    {
      "turn": "0.324919696",
      "x": "0.240910146",
      "y": "0.499952251"
    },
    {
      "turn": "0.324919696",
      "x": "0.907669990",
      "y": "0.499828694"
    },
    {
      "turn": "0.000000000",
      "x": "0.092330010",
      "y": "0.390292875"
    },
    {
      "turn": "0.324919696",
      "x": "0.573616877",
      "y": "0.390128749"
    },
    {
      "turn": "0.000000000",
      "x": "0.758276896",
      "y": "0.391288258"
    },
    {
      "turn": "0.324919696",
      "x": "0.243038496",
      "y": "0.283562921"
    },
    {
      "turn": "-0.324919696",
      "x": "0.907669990",
      "y": "0.282747822"
    },
    {
      "turn": "0.000000000",
      "x": "0.427698515",
      "y": "0.276805754"
    },
    {
      "turn": "-0.175425824",
      "x": "0.671514730",
      "y": "0.209563390"
    },
    {
      "turn": "0.015597918",
      "x": "0.103280087",
      "y": "0.157190215"
    },
    {
      "turn": "0.158384440",
      "x": "0.304514530",
      "y": "0.092330010"
    },
    {
      "turn": "0.141432327",
      "x": "0.518706273",
      "y": "0.094503167"
    },
    {
      "turn": "0.158384440",
      "x": "0.828016411",
      "y": "0.092330010"
    }
  ],
  "radius": "0.114126167"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 25,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "-0.158384440",
      "x": "0.304220615",
      "y": "0.907669990"
    },
    {
      "turn": "-0.146427724",
      "x": "0.519229945",
      "y": "0.906127624"
    },
    {
      "turn": "-0.158384440",
      "x": "0.828487975",
      "y": "0.907669990"
    },
    {
      "turn": "-0.000084734",
      "x": "0.094264873",
      "y": "0.851662728"
    },
    {
      "turn": "0.170385498",
      "x": "0.672532878",
      "y": "0.790819194"
    },
    {
      "turn": "0.000000000",
      "x": "0.427698515",
      "y": "0.724238889"
    },
    {
      "turn": "0.324919696",
      "x": "0.907669990",
      "y": "0.716909566"
    },
    {
      "turn": "-0.324919696",
      "x": "0.243038496",
      "y": "0.716341580"
    },
    {
      "turn": "-0.324919696",
      "x": "0.573616877",
      "y": "0.610915894"
    },
    {
      "turn": "0.000000000",
      "x": "0.092330010",
      "y": "0.609611626"
    },
    {
      "turn": "-0.000000000",
      "x": "0.758276896",
      "y": "0.608369130"
    },
    {
      "turn": "0.000000000",
      "x": "0.425570166",
      "y": "0.500522322"
    },
    {
      "turn": "0.324919696",
      "x": "0.240910146",
      "y": "0.499952251"
    },
    {
      "turn": "0.324919696",
      "x": "0.907669990",
      "y": "0.499828694"
    },
    {
      "turn": "0.000000000",
      "x": "0.092330010",
      "y": "0.390292875"
    },
    {
      "turn": "0.324919696",
      "x": "0.573616877",
      "y": "0.390128749"
    },
    {
      "turn": "0.000000000",
      "x": "0.758276896",
      "y": "0.391288258"
    },
    {
      "turn": "0.324919696",
      "x": "0.243038496",
      "y": "0.283562921"
    },
    {
      "turn": "-0.324919696",
      "x": "0.907669990",
      "y": "0.282747822"
    },
    {
      "turn": "0.000000000",
      "x": "0.427698515",
      "y": "0.276805754"
    },
    {
      "turn": "-0.175425824",
      "x": "0.671514730",
      "y": "0.209563390"
    },
    {
      "turn": "0.015597918",
      "x": "0.103280087",
      "y": "0.157190215"
    },
    {
      "turn": "0.158384440",
      "x": "0.304514530",
      "y": "0.092330010"
    },
    {
      "turn": "0.141432327",
      "x": "0.518706273",
      "y": "0.094503167"
    },
    {
      "turn": "0.158384440",
      "x": "0.828016411",
      "y": "0.092330010"
    }
  ],
  "radius": "0.114126167"
}

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?