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P94 · Packing and covering · Classic · Hard

Sum of radii in the regular pentagon · n = 29

Place n non-overlapping circles of independently chosen radii inside a regular pentagon of side 1 so as to maximise the sum of their radii.

Instancen = 29
ObjectiveMaximize the sum of the radii

Record comparison

the sum of the radii · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[3.68134, 3.68135)Richard Wang2026-10Erich Friedman ↗
Site record3.681347074zzzcyRecord details ↓

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

Formal definition

  • ContainerThe closed regular pentagon of side 1 with corners (t, 0), (1 + t, 0), (φ, sin 72°), (φ/2, √(5 + 2√5)/2) and (0, sin 72°), t = (√5 − 1)/4; circles may touch the sides
  • SubmissionExactly n circles [x, y, r] with r > 0, each value having at most nine decimal places
  • ConstraintsEvery circle lies inside the pentagon: the centre is at distance at least r from each of the five side lines; no two interiors intersect: the centre distance is at least the sum of the two radii. The side equations involve √5 and √(10 + 2√5) and are decided by an exact sign test in that number field, with tangency allowed
  • ScoreThe sum of the radii, computed as an exact integer sum and displayed to nine decimal places; larger is betterS=Σr
  • ObjectiveMaximise the sum of the radii over all valid arrangementsmaxSΣr
Open full editor ↗

Layout shown: site record

1.538841769y0
0x1.618033989
VERIFIED CONSTRUCTIONscored by the sum of the radii

Getting a feel for it

The incircle and five 108° corners

One circle can be at most the incircle, of radius cot(π/5)/2 ≈ 0.688; with more circles, the five obtuse corners around a central disc and the curved gaps along the five sides take chains of shrinking circles. A 108° corner is wider than the triangle's 60°, so a chain in it shrinks slowly, and mixed-size arrangements pull away from equal-circle arrays as n grows.

Literature records

Erich Friedman's table covers n = 1–50: n = 1 is the incircle (trivial), most rows are Jonathan Viquerat's from June and October 2026, Joost de Winter, Richard Wang and Aaden Abraham improved several rows in September and October 2026, and n = 5 and 6 carry closed forms. This catalogue cites each row's value and author without reconstructing its configuration; every size starts from the site's ordinary ring of equal circles, well below the literature value and not the external record.

Source
The record-holding arrangement for Sum of radii in the regular pentagon n = 29, 3.681347074
LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holderzzzcy
Solution methodAI · OpenAI: GPT-6 Astra
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 (7 changes)
  1. zzzcyAI · OpenAI: GPT-6 Astra
    3.681347073→3.681347074
  2. qiboqiAI · DeepSeek: DeepSeek V4.1 Flash 1017
    3.681347072→3.681347073
  3. zzzcyAI · OpenAI: GPT-6 Astra
    3.681347043→3.681347072
  4. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    3.681346999→3.681347043
  5. qiboqiAI · OpenAI: GPT-6 Luna
    3.681346484→3.681346999
  6. qiboqiAI · OpenAI: GPT-6 Luna
    3.658175047→3.681346484
  7. qiboqiAI · OpenAI: GPT-6 Luna
    1.557803324→3.658175047
ANSWER FORMAT

How to write your answer

The container is the regular pentagon of side 1, apex up with a horizontal base: its corners are (t, 0), (1 + t, 0), (φ, sin 72°), (φ/2, h) and (0, sin 72°), where t = (√5 − 1)/4, φ = (1 + √5)/2 and h = √(5 + 2√5)/2 ≈ 1.538842. Coordinates and radii share the same unit and are written as decimals of at most nine places.

Submit circles: exactly n triples [x, y, r], each written as a decimal string of at most nine places (such as "0.25"). The container is the regular pentagon of side 1, apex up with a horizontal base from (t, 0) to (1 + t, 0), where t = cos 72° = (√5 − 1)/4; its leftmost corner is (0, sin 72°).

The current leader's answer

{
  "circles": [
    [
      "0.474167158",
      "0.796991289",
      "0.160608479"
    ],
    [
      "0.802727304",
      "1.107871094",
      "0.123147500"
    ],
    [
      "1.201232757",
      "0.814219542",
      "0.123608903"
    ],
    [
      "1.052223753",
      "0.355646623",
      "0.123596820"
    ],
    [
      "0.557248815",
      "0.352353249",
      "0.123147972"
    ],
    [
      "1.005268270",
      "0.965896691",
      "0.124197549"
    ],
    [
      "1.141248054",
      "0.580208644",
      "0.117967720"
    ],
    [
      "0.804550757",
      "0.348143811",
      "0.124189791"
    ],
    [
      "0.391569730",
      "1.075269135",
      "0.129668793"
    ],
    [
      "1.029888882",
      "1.218352529",
      "0.129456003"
    ],
    [
      "1.380999541",
      "0.637435988",
      "0.128519083"
    ],
    [
      "0.935711797",
      "0.130187887",
      "0.130187887"
    ],
    [
      "0.313846311",
      "0.378991937",
      "0.121707901"
    ],
    [
      "0.590155571",
      "1.229391156",
      "0.121707287"
    ],
    [
      "1.239517271",
      "1.065126460",
      "0.130202021"
    ],
    [
      "1.301565090",
      "0.392975471",
      "0.128523280"
    ],
    [
      "0.676075148",
      "0.129449811",
      "0.129449811"
    ],
    [
      "0.243777520",
      "0.620406757",
      "0.129669804"
    ],
    [
      "0.769845502",
      "0.847447260",
      "0.139343997"
    ],
    [
      "0.683742851",
      "0.582375708",
      "0.139361213"
    ],
    [
      "0.177676118",
      "0.893326045",
      "0.151140333"
    ],
    [
      "0.809016995",
      "1.368639378",
      "0.137696626"
    ],
    [
      "1.452386077",
      "0.897234316",
      "0.140908474"
    ],
    [
      "1.206567403",
      "0.140878619",
      "0.140878619"
    ],
    [
      "0.409057985",
      "0.137694610",
      "0.137694610"
    ],
    [
      "0.923161664",
      "0.542357635",
      "0.103379004"
    ],
    [
      "0.987035122",
      "0.739027505",
      "0.103403143"
    ],
    [
      "0.606768541",
      "1.014333683",
      "0.093990898"
    ],
    [
      "0.453697717",
      "0.543213439",
      "0.093993553"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 29
}

The current leader's answer

{
  "circles": [
    [
      "0.474167158",
      "0.796991289",
      "0.160608479"
    ],
    [
      "0.802727304",
      "1.107871094",
      "0.123147500"
    ],
    [
      "1.201232757",
      "0.814219542",
      "0.123608903"
    ],
    [
      "1.052223753",
      "0.355646623",
      "0.123596820"
    ],
    [
      "0.557248815",
      "0.352353249",
      "0.123147972"
    ],
    [
      "1.005268270",
      "0.965896691",
      "0.124197549"
    ],
    [
      "1.141248054",
      "0.580208644",
      "0.117967720"
    ],
    [
      "0.804550757",
      "0.348143811",
      "0.124189791"
    ],
    [
      "0.391569730",
      "1.075269135",
      "0.129668793"
    ],
    [
      "1.029888882",
      "1.218352529",
      "0.129456003"
    ],
    [
      "1.380999541",
      "0.637435988",
      "0.128519083"
    ],
    [
      "0.935711797",
      "0.130187887",
      "0.130187887"
    ],
    [
      "0.313846311",
      "0.378991937",
      "0.121707901"
    ],
    [
      "0.590155571",
      "1.229391156",
      "0.121707287"
    ],
    [
      "1.239517271",
      "1.065126460",
      "0.130202021"
    ],
    [
      "1.301565090",
      "0.392975471",
      "0.128523280"
    ],
    [
      "0.676075148",
      "0.129449811",
      "0.129449811"
    ],
    [
      "0.243777520",
      "0.620406757",
      "0.129669804"
    ],
    [
      "0.769845502",
      "0.847447260",
      "0.139343997"
    ],
    [
      "0.683742851",
      "0.582375708",
      "0.139361213"
    ],
    [
      "0.177676118",
      "0.893326045",
      "0.151140333"
    ],
    [
      "0.809016995",
      "1.368639378",
      "0.137696626"
    ],
    [
      "1.452386077",
      "0.897234316",
      "0.140908474"
    ],
    [
      "1.206567403",
      "0.140878619",
      "0.140878619"
    ],
    [
      "0.409057985",
      "0.137694610",
      "0.137694610"
    ],
    [
      "0.923161664",
      "0.542357635",
      "0.103379004"
    ],
    [
      "0.987035122",
      "0.739027505",
      "0.103403143"
    ],
    [
      "0.606768541",
      "1.014333683",
      "0.093990898"
    ],
    [
      "0.453697717",
      "0.543213439",
      "0.093993553"
    ]
  ]
}

Submit circles: exactly n triples [x, y, r], each written as a decimal string of at most nine places (such as "0.25"). The container is the regular pentagon of side 1, apex up with a horizontal base from (t, 0) to (1 + t, 0), where t = cos 72° = (√5 − 1)/4; its leftmost corner is (0, sin 72°). · Verifier v1.0.0

DISCUSSION

Discussion

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33 34 43 50ai说是新构型 应该可以更好优化zzzcy#3081 replies1017的数学能力怎么样zzzcy#3081 replies