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

Regular pentagons in a square · n = 9

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

Instancen = 9
ObjectiveMinimize container square side

Record comparison

container square side · Lower is better
RecordValue / intervalAuthor / holderSource
External best known[4.60032, 4.60033)Yanis Gomes2026-10Erich Friedman ↗
Site record4.600329539condorRecord details ↓

Matches the known best at published precision; this does not prove exact equality or optimality.

Formal definition

  • ContainerThe editor and certificate normalise the container to the unit square [0,1]², while public scores are reported in the literature's unit-piece scale. Boundary contact is allowed.
  • SubmissionSubmit {radius, placements:[{x,y,turn},…]}, with each value written as a decimal string of at most nine places. Here x, y are centre coordinates; circles require turn = 0 and polygons use turn = tan(θ/2). Coordinates and turn must lie in [−4, 4], with 0 < radius ≤ 4.
  • ConstraintsExactly n congruent pieces lie wholly inside the container with pairwise disjoint interiors. Polygons may rotate independently by arbitrary angles.
  • PrecisionInput decimals are parsed as exact rationals and regular-polygon vertices are defined algebraically, with zero floating-point tolerance in feasibility checks.
  • ObjectiveContainer square side = 1/(2 sin(π/5) r), where smaller is better. Pages, leaderboards, and literature records share this unit with automatic conversion.
Open full editor ↗

Layout shown: site record

VERIFIED CONSTRUCTIONVERIFIED CONSTRUCTION

Getting a feel for it

Literature targets and coordinate certificates

Literature targets are converted at their published precision, while nine-decimal coordinate certificates are stored and displayed alongside the source values.

Source
The record-holding arrangement for Regular pentagons in a square n = 9, 4.600329539
LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holdercondor
Solution methodAI · OpenAI: GPT-6.1 Terra
Challenge this record ↗Submit a proof / idea ↓Share a proof or bound argument in the discussion board. Accepted contributions earn proof points.
Record holder's solver noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

Record history (1 changes)
  1. condorAI · OpenAI: GPT-6.1 Terra
    4.600330011→4.600329539
ANSWER FORMAT

How to write your answer

the unit square [0,1]²

Submit {radius, placements:[{x,y,turn},…]}, with each value written as a decimal string of at most nine places. Here x, y are centre coordinates; circles require turn = 0 and polygons use turn = tan(θ/2). Coordinates and turn must lie in [−4, 4], with 0 < radius ≤ 4.

The current leader's answer

{
  "placements": [
    {
      "turn": "0.078701707",
      "x": "0.164756767",
      "y": "0.164756767"
    },
    {
      "turn": "-0.240078759",
      "x": "0.509656880",
      "y": "0.182634283"
    },
    {
      "turn": "0.078701707",
      "x": "0.817365717",
      "y": "0.164756767"
    },
    {
      "turn": "-0.240078759",
      "x": "0.182634283",
      "y": "0.509656880"
    },
    {
      "turn": "0.078701707",
      "x": "0.488851091",
      "y": "0.488851091"
    },
    {
      "turn": "-0.240078759",
      "x": "0.835243233",
      "y": "0.509656880"
    },
    {
      "turn": "0.078701707",
      "x": "0.164756767",
      "y": "0.817365717"
    },
    {
      "turn": "-0.240078759",
      "x": "0.509656880",
      "y": "0.835243233"
    },
    {
      "turn": "0.082803651",
      "x": "0.817289275",
      "y": "0.816858969"
    }
  ],
  "radius": "0.184910842"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 9,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.078701707",
      "x": "0.164756767",
      "y": "0.164756767"
    },
    {
      "turn": "-0.240078759",
      "x": "0.509656880",
      "y": "0.182634283"
    },
    {
      "turn": "0.078701707",
      "x": "0.817365717",
      "y": "0.164756767"
    },
    {
      "turn": "-0.240078759",
      "x": "0.182634283",
      "y": "0.509656880"
    },
    {
      "turn": "0.078701707",
      "x": "0.488851091",
      "y": "0.488851091"
    },
    {
      "turn": "-0.240078759",
      "x": "0.835243233",
      "y": "0.509656880"
    },
    {
      "turn": "0.078701707",
      "x": "0.164756767",
      "y": "0.817365717"
    },
    {
      "turn": "-0.240078759",
      "x": "0.509656880",
      "y": "0.835243233"
    },
    {
      "turn": "0.082803651",
      "x": "0.817289275",
      "y": "0.816858969"
    }
  ],
  "radius": "0.184910842"
}

Submit {radius, placements:[{x,y,turn},…]}, with each value written as a decimal string of at most nine places. Here x, y are centre coordinates; circles require turn = 0 and polygons use turn = tan(θ/2). Coordinates and turn must lie in [−4, 4], with 0 < radius ≤ 4. · Verifier v1.0.0

DISCUSSION

Discussion

Discuss strategies, share construction methods, or ask questions. Any signed-in user can post under their public byline and unique #UID. New posts appear after moderation.

外部记录更新NUE_13#311 replies