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P86 · Packing and covering · Classic · Hard

Regular pentagons in a square · n = 10

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

Instancen = 10
ObjectiveMinimize container square side

Record comparison

container square side · Lower is better
RecordValue / intervalAuthor / holderSource
External best known[4.90581, 4.90582)Jonathan ViqueratErich Friedman ↗
Site record4.905820411邱仲普Record details ↓

The site record has not been established to match or exceed this reference.

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 = 10, 4.905820411
LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holder邱仲普
Solution methodAI · OpenAI: GPT-5.6 Sol
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 (22 changes)
  1. 邱仲普AI · OpenAI: GPT-5.6 Sol
    4.905820439→4.905820411
  2. zzzcyAI · OpenAI: GPT-6 Astra
    4.914230701→4.905820439
  3. NUE_13Human
    4.914332594→4.914230701
  4. 今天Human
    4.914336313→4.914332594
  5. NUE_13Human
    4.914336994→4.914336313
  6. hadsajncHuman
    4.914355051→4.914336994
  7. NUE_13Human
    4.914827779→4.914355051
  8. NUE_13Human
    4.915910863→4.914827779
  9. NUE_13Human
    4.917387554→4.915910863
  10. NUE_13Human
    4.919600812→4.917387554
  11. NUE_13Human
    4.923169993→4.919600812
  12. NUE_13Human
    4.924103572→4.923169993
  13. NUE_13Human
    4.926782307→4.924103572
  14. NUE_13Human
    4.963925797→4.926782307
  15. NUE_13AI · MiniMax: MiniMax M3
    4.967560999→4.963925797
  16. NUE_13Human
    5.124901799→4.967560999
  17. NUE_13Human
    5.127202376→5.124901799
  18. NUE_13Human
    5.128031315→5.127202376
  19. NUE_13Human
    5.129601435→5.128031315
  20. qiboqiHuman
    5.129601497→5.129601435
  21. 玄月Human
    5.316567553→5.129601497
  22. XSsMCHuman
    9.357158883→5.316567553
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": "-0.000000002",
      "x": "0.593042595",
      "y": "0.673649227"
    },
    {
      "turn": "0.324919697",
      "x": "0.859719495",
      "y": "0.530821119"
    },
    {
      "turn": "0.000000000",
      "x": "0.645852148",
      "y": "0.347865371"
    },
    {
      "turn": "0.158384440",
      "x": "0.412347747",
      "y": "0.140280505"
    },
    {
      "turn": "0.324919695",
      "x": "0.859719495",
      "y": "0.164909623"
    },
    {
      "turn": "0.324919696",
      "x": "0.365291139",
      "y": "0.452650859"
    },
    {
      "turn": "-0.158384440",
      "x": "0.835090376",
      "y": "0.859719495"
    },
    {
      "turn": "-0.158384440",
      "x": "0.329925889",
      "y": "0.859719495"
    },
    {
      "turn": "0.000000000",
      "x": "0.140280505",
      "y": "0.620269188"
    },
    {
      "turn": "0.000000000",
      "x": "0.140280505",
      "y": "0.285032530"
    }
  ],
  "radius": "0.173396239"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 10,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "-0.000000002",
      "x": "0.593042595",
      "y": "0.673649227"
    },
    {
      "turn": "0.324919697",
      "x": "0.859719495",
      "y": "0.530821119"
    },
    {
      "turn": "0.000000000",
      "x": "0.645852148",
      "y": "0.347865371"
    },
    {
      "turn": "0.158384440",
      "x": "0.412347747",
      "y": "0.140280505"
    },
    {
      "turn": "0.324919695",
      "x": "0.859719495",
      "y": "0.164909623"
    },
    {
      "turn": "0.324919696",
      "x": "0.365291139",
      "y": "0.452650859"
    },
    {
      "turn": "-0.158384440",
      "x": "0.835090376",
      "y": "0.859719495"
    },
    {
      "turn": "-0.158384440",
      "x": "0.329925889",
      "y": "0.859719495"
    },
    {
      "turn": "0.000000000",
      "x": "0.140280505",
      "y": "0.620269188"
    },
    {
      "turn": "0.000000000",
      "x": "0.140280505",
      "y": "0.285032530"
    }
  ],
  "radius": "0.173396239"
}

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

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外部记录更新NUE_13#311 replies