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P94 · Packing and covering · Classic · Hard

Sum of radii in the regular pentagon · n = 37

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 = 37
ObjectiveMaximize the sum of the radii

Record comparison

the sum of the radii · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[4.14946, 4.14947)Joost de Winter2026-10Erich Friedman ↗
Site record4.150872851zzzcyRecord details ↓

Numerically exceeds the published range · awaiting review.

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 = 37, 4.150872851
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 (10 changes)
  1. zzzcyAI · OpenAI: GPT-6 Astra
    4.150872849→4.150872851
  2. qiboqiAI · DeepSeek: DeepSeek V4.1 Flash 1017
    4.150872848→4.150872849
  3. qiboqiAI · DeepSeek: DeepSeek V4.1 Flash 1017
    4.150872813→4.150872848
  4. qiboqiAI · DeepSeek: DeepSeek V4.1 Flash 1017
    4.149460253→4.150872813
  5. zzzcyAI · OpenAI: GPT-6 Astra
    4.147488101→4.149460253
  6. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    4.147488003→4.147488101
  7. qiboqiAI · OpenAI: GPT-6 Luna
    4.146759997→4.147488003
  8. qiboqiAI · OpenAI: GPT-6 Luna
    4.137092471→4.146759997
  9. qiboqiAI · OpenAI: GPT-6 Luna
    4.137091818→4.137092471
  10. qiboqiAI · OpenAI: GPT-6 Luna
    1.592483034→4.137091818
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": [
    [
      "0.926295775",
      "0.891600068",
      "0.075951317"
    ],
    [
      "0.693578609",
      "1.122732721",
      "0.098646392"
    ],
    [
      "0.752883152",
      "0.931142429",
      "0.101912496"
    ],
    [
      "1.167976685",
      "0.564720028",
      "0.116446070"
    ],
    [
      "0.706709223",
      "0.558691726",
      "0.116171731"
    ],
    [
      "0.937341128",
      "0.551338296",
      "0.114577372"
    ],
    [
      "1.472909378",
      "0.730960681",
      "0.070008353"
    ],
    [
      "0.567983225",
      "0.359749757",
      "0.126362409"
    ],
    [
      "1.207493516",
      "0.135688524",
      "0.135688524"
    ],
    [
      "1.387901479",
      "0.579182955",
      "0.103953772"
    ],
    [
      "0.326814170",
      "1.039664519",
      "0.120411169"
    ],
    [
      "1.039835221",
      "0.747953208",
      "0.107148726"
    ],
    [
      "0.829457385",
      "0.740693329",
      "0.103354337"
    ],
    [
      "0.809016994",
      "1.418591221",
      "0.097284736"
    ],
    [
      "0.817023201",
      "0.346442698",
      "0.123032833"
    ],
    [
      "0.546990530",
      "0.975948659",
      "0.108799084"
    ],
    [
      "1.099456385",
      "0.955226129",
      "0.108528688"
    ],
    [
      "0.392141762",
      "0.810817127",
      "0.117577954"
    ],
    [
      "1.277469695",
      "0.790198521",
      "0.134211605"
    ],
    [
      "0.472029397",
      "0.586867959",
      "0.120193496"
    ],
    [
      "1.062806530",
      "0.349858241",
      "0.122774223"
    ],
    [
      "1.306469085",
      "1.036620324",
      "0.113910681"
    ],
    [
      "1.489691916",
      "0.909355650",
      "0.109174287"
    ],
    [
      "0.499563442",
      "1.184331429",
      "0.104912655"
    ],
    [
      "0.616661803",
      "0.767371101",
      "0.111106995"
    ],
    [
      "1.304143833",
      "0.371943466",
      "0.119571503"
    ],
    [
      "0.652564380",
      "1.310065352",
      "0.093123469"
    ],
    [
      "0.966682066",
      "1.302600067",
      "0.098450347"
    ],
    [
      "0.805462632",
      "1.250305701",
      "0.071038315"
    ],
    [
      "1.128203943",
      "1.171458309",
      "0.109606081"
    ],
    [
      "0.911693730",
      "1.089064212",
      "0.122051985"
    ],
    [
      "0.315778228",
      "0.393854624",
      "0.128138087"
    ],
    [
      "0.233115082",
      "0.630686946",
      "0.122705976"
    ],
    [
      "0.943229610",
      "0.128668604",
      "0.128668604"
    ],
    [
      "0.157538501",
      "0.864597113",
      "0.123110593"
    ],
    [
      "0.684023963",
      "0.130543824",
      "0.130543824"
    ],
    [
      "0.411985627",
      "0.141724162",
      "0.141724162"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 37
}

The current leader's answer

{
  "circles": [
    [
      "0.926295775",
      "0.891600068",
      "0.075951317"
    ],
    [
      "0.693578609",
      "1.122732721",
      "0.098646392"
    ],
    [
      "0.752883152",
      "0.931142429",
      "0.101912496"
    ],
    [
      "1.167976685",
      "0.564720028",
      "0.116446070"
    ],
    [
      "0.706709223",
      "0.558691726",
      "0.116171731"
    ],
    [
      "0.937341128",
      "0.551338296",
      "0.114577372"
    ],
    [
      "1.472909378",
      "0.730960681",
      "0.070008353"
    ],
    [
      "0.567983225",
      "0.359749757",
      "0.126362409"
    ],
    [
      "1.207493516",
      "0.135688524",
      "0.135688524"
    ],
    [
      "1.387901479",
      "0.579182955",
      "0.103953772"
    ],
    [
      "0.326814170",
      "1.039664519",
      "0.120411169"
    ],
    [
      "1.039835221",
      "0.747953208",
      "0.107148726"
    ],
    [
      "0.829457385",
      "0.740693329",
      "0.103354337"
    ],
    [
      "0.809016994",
      "1.418591221",
      "0.097284736"
    ],
    [
      "0.817023201",
      "0.346442698",
      "0.123032833"
    ],
    [
      "0.546990530",
      "0.975948659",
      "0.108799084"
    ],
    [
      "1.099456385",
      "0.955226129",
      "0.108528688"
    ],
    [
      "0.392141762",
      "0.810817127",
      "0.117577954"
    ],
    [
      "1.277469695",
      "0.790198521",
      "0.134211605"
    ],
    [
      "0.472029397",
      "0.586867959",
      "0.120193496"
    ],
    [
      "1.062806530",
      "0.349858241",
      "0.122774223"
    ],
    [
      "1.306469085",
      "1.036620324",
      "0.113910681"
    ],
    [
      "1.489691916",
      "0.909355650",
      "0.109174287"
    ],
    [
      "0.499563442",
      "1.184331429",
      "0.104912655"
    ],
    [
      "0.616661803",
      "0.767371101",
      "0.111106995"
    ],
    [
      "1.304143833",
      "0.371943466",
      "0.119571503"
    ],
    [
      "0.652564380",
      "1.310065352",
      "0.093123469"
    ],
    [
      "0.966682066",
      "1.302600067",
      "0.098450347"
    ],
    [
      "0.805462632",
      "1.250305701",
      "0.071038315"
    ],
    [
      "1.128203943",
      "1.171458309",
      "0.109606081"
    ],
    [
      "0.911693730",
      "1.089064212",
      "0.122051985"
    ],
    [
      "0.315778228",
      "0.393854624",
      "0.128138087"
    ],
    [
      "0.233115082",
      "0.630686946",
      "0.122705976"
    ],
    [
      "0.943229610",
      "0.128668604",
      "0.128668604"
    ],
    [
      "0.157538501",
      "0.864597113",
      "0.123110593"
    ],
    [
      "0.684023963",
      "0.130543824",
      "0.130543824"
    ],
    [
      "0.411985627",
      "0.141724162",
      "0.141724162"
    ]
  ]
}

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