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P94 · Packing and covering · Classic · Hard

Sum of radii in the regular pentagon · n = 38

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 = 38
ObjectiveMaximize the sum of the radii

Record comparison

the sum of the radii · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[4.21143, 4.21144)Richard Wang2026-10Erich Friedman ↗
Site record4.21143237zzzcyRecord 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 = 38, 4.21143237
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 (8 changes)
  1. zzzcyAI · OpenAI: GPT-6 Astra
    4.211432368→4.21143237
  2. qiboqiAI · DeepSeek: DeepSeek V4.1 Flash 1017
    4.211432367→4.211432368
  3. zzzcyAI · OpenAI: GPT-6 Astra
    4.206799817→4.211432367
  4. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    4.206799728→4.206799817
  5. qiboqiAI · OpenAI: GPT-6 Luna
    4.200807105→4.206799728
  6. qiboqiAI · OpenAI: GPT-6 Luna
    4.200806435→4.200807105
  7. qiboqiAI · OpenAI: GPT-6 Luna
    4.181464703→4.200806435
  8. qiboqiAI · OpenAI: GPT-6 Luna
    1.595857994→4.181464703
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": [
    [
      "0.585199965",
      "1.216330427",
      "0.129360807"
    ],
    [
      "1.025004359",
      "1.231840747",
      "0.121414856"
    ],
    [
      "1.224082247",
      "0.643511876",
      "0.103325096"
    ],
    [
      "1.372669158",
      "0.509357666",
      "0.096863370"
    ],
    [
      "1.003576211",
      "0.101856104",
      "0.101856104"
    ],
    [
      "0.723247111",
      "0.279633851",
      "0.103325096"
    ],
    [
      "1.375687603",
      "1.041369946",
      "0.069382531"
    ],
    [
      "1.492951383",
      "0.910414712",
      "0.106401619"
    ],
    [
      "1.223089114",
      "0.118269580",
      "0.118269580"
    ],
    [
      "1.080525056",
      "0.314492173",
      "0.124274880"
    ],
    [
      "0.386322297",
      "0.106401620",
      "0.106401620"
    ],
    [
      "0.600534277",
      "0.107815022",
      "0.107815022"
    ],
    [
      "0.896327043",
      "0.241188683",
      "0.073973221"
    ],
    [
      "0.877356284",
      "0.417793757",
      "0.103647841"
    ],
    [
      "0.225231556",
      "0.650771796",
      "0.121414857"
    ],
    [
      "1.306532475",
      "0.321966714",
      "0.101856104"
    ],
    [
      "1.045062014",
      "0.539639102",
      "0.103647841"
    ],
    [
      "1.207161134",
      "0.467022870",
      "0.073973221"
    ],
    [
      "1.255325220",
      "0.878168683",
      "0.133402467"
    ],
    [
      "0.849933346",
      "0.631874434",
      "0.112182077"
    ],
    [
      "1.425412015",
      "0.707123779",
      "0.107815022"
    ],
    [
      "0.671921026",
      "0.491405420",
      "0.114577552"
    ],
    [
      "0.278817455",
      "0.431258151",
      "0.104544628"
    ],
    [
      "1.038536150",
      "0.757766898",
      "0.114577552"
    ],
    [
      "0.637283462",
      "0.722291066",
      "0.118891808"
    ],
    [
      "0.829654449",
      "0.862056769",
      "0.118891808"
    ],
    [
      "0.813813587",
      "1.105245612",
      "0.124812410"
    ],
    [
      "0.804919466",
      "0.096863370",
      "0.096863370"
    ],
    [
      "0.375889782",
      "1.064257677",
      "0.129360807"
    ],
    [
      "0.452578770",
      "0.572132951",
      "0.119148697"
    ],
    [
      "0.165019023",
      "0.901862504",
      "0.136794540"
    ],
    [
      "0.608784195",
      "0.963787767",
      "0.124280692"
    ],
    [
      "0.410892223",
      "0.812506106",
      "0.124812410"
    ],
    [
      "0.804809603",
      "1.366697568",
      "0.136794541"
    ],
    [
      "0.490420618",
      "0.322432960",
      "0.133402467"
    ],
    [
      "1.029540191",
      "0.991319959",
      "0.119148697"
    ],
    [
      "1.217215287",
      "1.113044082",
      "0.104544628"
    ],
    [
      "0.298012969",
      "0.258393496",
      "0.069382531"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 38
}

The current leader's answer

{
  "circles": [
    [
      "0.585199965",
      "1.216330427",
      "0.129360807"
    ],
    [
      "1.025004359",
      "1.231840747",
      "0.121414856"
    ],
    [
      "1.224082247",
      "0.643511876",
      "0.103325096"
    ],
    [
      "1.372669158",
      "0.509357666",
      "0.096863370"
    ],
    [
      "1.003576211",
      "0.101856104",
      "0.101856104"
    ],
    [
      "0.723247111",
      "0.279633851",
      "0.103325096"
    ],
    [
      "1.375687603",
      "1.041369946",
      "0.069382531"
    ],
    [
      "1.492951383",
      "0.910414712",
      "0.106401619"
    ],
    [
      "1.223089114",
      "0.118269580",
      "0.118269580"
    ],
    [
      "1.080525056",
      "0.314492173",
      "0.124274880"
    ],
    [
      "0.386322297",
      "0.106401620",
      "0.106401620"
    ],
    [
      "0.600534277",
      "0.107815022",
      "0.107815022"
    ],
    [
      "0.896327043",
      "0.241188683",
      "0.073973221"
    ],
    [
      "0.877356284",
      "0.417793757",
      "0.103647841"
    ],
    [
      "0.225231556",
      "0.650771796",
      "0.121414857"
    ],
    [
      "1.306532475",
      "0.321966714",
      "0.101856104"
    ],
    [
      "1.045062014",
      "0.539639102",
      "0.103647841"
    ],
    [
      "1.207161134",
      "0.467022870",
      "0.073973221"
    ],
    [
      "1.255325220",
      "0.878168683",
      "0.133402467"
    ],
    [
      "0.849933346",
      "0.631874434",
      "0.112182077"
    ],
    [
      "1.425412015",
      "0.707123779",
      "0.107815022"
    ],
    [
      "0.671921026",
      "0.491405420",
      "0.114577552"
    ],
    [
      "0.278817455",
      "0.431258151",
      "0.104544628"
    ],
    [
      "1.038536150",
      "0.757766898",
      "0.114577552"
    ],
    [
      "0.637283462",
      "0.722291066",
      "0.118891808"
    ],
    [
      "0.829654449",
      "0.862056769",
      "0.118891808"
    ],
    [
      "0.813813587",
      "1.105245612",
      "0.124812410"
    ],
    [
      "0.804919466",
      "0.096863370",
      "0.096863370"
    ],
    [
      "0.375889782",
      "1.064257677",
      "0.129360807"
    ],
    [
      "0.452578770",
      "0.572132951",
      "0.119148697"
    ],
    [
      "0.165019023",
      "0.901862504",
      "0.136794540"
    ],
    [
      "0.608784195",
      "0.963787767",
      "0.124280692"
    ],
    [
      "0.410892223",
      "0.812506106",
      "0.124812410"
    ],
    [
      "0.804809603",
      "1.366697568",
      "0.136794541"
    ],
    [
      "0.490420618",
      "0.322432960",
      "0.133402467"
    ],
    [
      "1.029540191",
      "0.991319959",
      "0.119148697"
    ],
    [
      "1.217215287",
      "1.113044082",
      "0.104544628"
    ],
    [
      "0.298012969",
      "0.258393496",
      "0.069382531"
    ]
  ]
}

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