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P94 · Packing and covering · Classic · Hard

Sum of radii in the regular pentagon · n = 42

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 = 42
ObjectiveMaximize the sum of the radii

Record comparison

the sum of the radii · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[4.44142, 4.44143)Jonathan Viquerat2026-10Erich Friedman ↗
Site record4.441416952zzzcyRecord details ↓

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

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 = 42, 4.441416952
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
    4.441416951→4.441416952
  2. qiboqiAI · DeepSeek: DeepSeek V4.1 Flash 1017
    4.441416947→4.441416951
  3. zzzcyAI · OpenAI: GPT-6 Astra
    4.440485719→4.441416947
  4. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    4.440485611→4.440485719
  5. qiboqiAI · OpenAI: GPT-6 Luna
    4.42916054→4.440485611
  6. qiboqiAI · OpenAI: GPT-6 Luna
    4.41525673→4.42916054
  7. qiboqiAI · OpenAI: GPT-6 Luna
    1.607845638→4.41525673
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": [
    [
      "0.628386301",
      "1.287211644",
      "0.097400989"
    ],
    [
      "0.992758447",
      "1.270357763",
      "0.109207607"
    ],
    [
      "0.413972633",
      "0.447580920",
      "0.074778408"
    ],
    [
      "1.197651023",
      "0.699399246",
      "0.130754232"
    ],
    [
      "1.354607949",
      "0.510917714",
      "0.114522681"
    ],
    [
      "1.033844217",
      "0.097760385",
      "0.097760385"
    ],
    [
      "1.337952421",
      "1.033709205",
      "0.097760385"
    ],
    [
      "0.310788171",
      "0.309745105",
      "0.097400989"
    ],
    [
      "1.497194219",
      "0.911793297",
      "0.102792446"
    ],
    [
      "1.168682473",
      "1.146763412",
      "0.105791989"
    ],
    [
      "1.234333910",
      "0.102792447",
      "0.102792447"
    ],
    [
      "1.126053878",
      "0.244551649",
      "0.075589829"
    ],
    [
      "0.615477891",
      "0.109207607",
      "0.109207607"
    ],
    [
      "0.197406460",
      "0.688555561",
      "0.106627443"
    ],
    [
      "0.809016994",
      "1.397582774",
      "0.114280927"
    ],
    [
      "1.297005171",
      "0.299992451",
      "0.104126688"
    ],
    [
      "1.116840464",
      "0.450689916",
      "0.130754233"
    ],
    [
      "1.158938436",
      "0.934088278",
      "0.107106244"
    ],
    [
      "1.431985082",
      "0.715417900",
      "0.104126688"
    ],
    [
      "0.613149339",
      "0.516662552",
      "0.136038176"
    ],
    [
      "0.249557454",
      "0.492973763",
      "0.095787887"
    ],
    [
      "0.973908773",
      "1.050473225",
      "0.111483399"
    ],
    [
      "0.969415482",
      "0.829839212",
      "0.109196363"
    ],
    [
      "0.570522642",
      "0.765682474",
      "0.116603776"
    ],
    [
      "0.751378601",
      "0.942088478",
      "0.136038176"
    ],
    [
      "0.799178062",
      "1.178432039",
      "0.105090559"
    ],
    [
      "0.830450355",
      "0.105791990",
      "0.105791990"
    ],
    [
      "0.313999371",
      "1.047391645",
      "0.106627443"
    ],
    [
      "0.399052267",
      "0.626181990",
      "0.104444803"
    ],
    [
      "0.139126704",
      "0.905851510",
      "0.118348243"
    ],
    [
      "0.513796520",
      "0.979328494",
      "0.104444803"
    ],
    [
      "0.471150508",
      "1.174967058",
      "0.095787886"
    ],
    [
      "0.355654216",
      "0.835497457",
      "0.109322256"
    ],
    [
      "0.512900654",
      "0.297360775",
      "0.105090559"
    ],
    [
      "0.392046948",
      "0.114280927",
      "0.114280927"
    ],
    [
      "1.326269865",
      "0.860753098",
      "0.075589830"
    ],
    [
      "0.855523345",
      "0.479315252",
      "0.109196363"
    ],
    [
      "0.790877803",
      "0.694084741",
      "0.115091352"
    ],
    [
      "0.986893640",
      "0.630395335",
      "0.091011893"
    ],
    [
      "0.729473073",
      "0.298177497",
      "0.111483399"
    ],
    [
      "0.630846425",
      "1.115049823",
      "0.074778408"
    ],
    [
      "0.947574571",
      "0.283577187",
      "0.107106244"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 42
}

The current leader's answer

{
  "circles": [
    [
      "0.628386301",
      "1.287211644",
      "0.097400989"
    ],
    [
      "0.992758447",
      "1.270357763",
      "0.109207607"
    ],
    [
      "0.413972633",
      "0.447580920",
      "0.074778408"
    ],
    [
      "1.197651023",
      "0.699399246",
      "0.130754232"
    ],
    [
      "1.354607949",
      "0.510917714",
      "0.114522681"
    ],
    [
      "1.033844217",
      "0.097760385",
      "0.097760385"
    ],
    [
      "1.337952421",
      "1.033709205",
      "0.097760385"
    ],
    [
      "0.310788171",
      "0.309745105",
      "0.097400989"
    ],
    [
      "1.497194219",
      "0.911793297",
      "0.102792446"
    ],
    [
      "1.168682473",
      "1.146763412",
      "0.105791989"
    ],
    [
      "1.234333910",
      "0.102792447",
      "0.102792447"
    ],
    [
      "1.126053878",
      "0.244551649",
      "0.075589829"
    ],
    [
      "0.615477891",
      "0.109207607",
      "0.109207607"
    ],
    [
      "0.197406460",
      "0.688555561",
      "0.106627443"
    ],
    [
      "0.809016994",
      "1.397582774",
      "0.114280927"
    ],
    [
      "1.297005171",
      "0.299992451",
      "0.104126688"
    ],
    [
      "1.116840464",
      "0.450689916",
      "0.130754233"
    ],
    [
      "1.158938436",
      "0.934088278",
      "0.107106244"
    ],
    [
      "1.431985082",
      "0.715417900",
      "0.104126688"
    ],
    [
      "0.613149339",
      "0.516662552",
      "0.136038176"
    ],
    [
      "0.249557454",
      "0.492973763",
      "0.095787887"
    ],
    [
      "0.973908773",
      "1.050473225",
      "0.111483399"
    ],
    [
      "0.969415482",
      "0.829839212",
      "0.109196363"
    ],
    [
      "0.570522642",
      "0.765682474",
      "0.116603776"
    ],
    [
      "0.751378601",
      "0.942088478",
      "0.136038176"
    ],
    [
      "0.799178062",
      "1.178432039",
      "0.105090559"
    ],
    [
      "0.830450355",
      "0.105791990",
      "0.105791990"
    ],
    [
      "0.313999371",
      "1.047391645",
      "0.106627443"
    ],
    [
      "0.399052267",
      "0.626181990",
      "0.104444803"
    ],
    [
      "0.139126704",
      "0.905851510",
      "0.118348243"
    ],
    [
      "0.513796520",
      "0.979328494",
      "0.104444803"
    ],
    [
      "0.471150508",
      "1.174967058",
      "0.095787886"
    ],
    [
      "0.355654216",
      "0.835497457",
      "0.109322256"
    ],
    [
      "0.512900654",
      "0.297360775",
      "0.105090559"
    ],
    [
      "0.392046948",
      "0.114280927",
      "0.114280927"
    ],
    [
      "1.326269865",
      "0.860753098",
      "0.075589830"
    ],
    [
      "0.855523345",
      "0.479315252",
      "0.109196363"
    ],
    [
      "0.790877803",
      "0.694084741",
      "0.115091352"
    ],
    [
      "0.986893640",
      "0.630395335",
      "0.091011893"
    ],
    [
      "0.729473073",
      "0.298177497",
      "0.111483399"
    ],
    [
      "0.630846425",
      "1.115049823",
      "0.074778408"
    ],
    [
      "0.947574571",
      "0.283577187",
      "0.107106244"
    ]
  ]
}

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