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P94 · Packing and covering · Classic · Hard

Sum of radii in the regular pentagon · n = 46

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 = 46
ObjectiveMaximize the sum of the radii

Record comparison

the sum of the radii · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[4.66314, 4.66315)Jonathan Viquerat2026-10Erich Friedman ↗
Site record4.663140405zzzcyRecord 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 = 46, 4.663140405
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
    4.663140404→4.663140405
  2. qiboqiAI · DeepSeek: DeepSeek V4.1 Flash 1017
    4.663140403→4.663140404
  3. zzzcyAI · OpenAI: GPT-6 Astra
    4.660954884→4.663140403
  4. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    4.6609548→4.660954884
  5. qiboqiAI · OpenAI: GPT-6 Luna
    4.654063858→4.6609548
  6. qiboqiAI · OpenAI: GPT-6 Luna
    4.646332538→4.654063858
  7. qiboqiAI · OpenAI: GPT-6 Luna
    1.617861492→4.646332538
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": [
    [
      "0.630078780",
      "1.281649381",
      "0.102895768"
    ],
    [
      "0.315426830",
      "0.313250256",
      "0.102895768"
    ],
    [
      "1.232463736",
      "0.654568074",
      "0.098874975"
    ],
    [
      "1.367303727",
      "0.506792603",
      "0.101173550"
    ],
    [
      "1.013079046",
      "0.102895767",
      "0.102895767"
    ],
    [
      "0.615646019",
      "0.422038644",
      "0.075843633"
    ],
    [
      "1.318134546",
      "1.041760052",
      "0.102895767"
    ],
    [
      "0.446678348",
      "0.466496696",
      "0.098874974"
    ],
    [
      "1.486466109",
      "0.908307520",
      "0.111918323"
    ],
    [
      "1.227703572",
      "0.111918324",
      "0.111918324"
    ],
    [
      "1.104443685",
      "0.281571003",
      "0.097783920"
    ],
    [
      "0.390330417",
      "0.111918324",
      "0.111918324"
    ],
    [
      "0.604954944",
      "0.102895767",
      "0.102895767"
    ],
    [
      "0.907891970",
      "0.275079133",
      "0.098874975"
    ],
    [
      "1.002387971",
      "0.422038644",
      "0.075843632"
    ],
    [
      "0.189309546",
      "0.701399343",
      "0.102895768"
    ],
    [
      "0.809016996",
      "1.400503111",
      "0.111918324"
    ],
    [
      "1.302607159",
      "0.313250255",
      "0.102895767"
    ],
    [
      "1.011036583",
      "0.622550816",
      "0.124854971"
    ],
    [
      "1.171355641",
      "0.466496695",
      "0.098874975"
    ],
    [
      "1.131829570",
      "0.964288195",
      "0.098874974"
    ],
    [
      "0.809016995",
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      "0.087560994"
    ],
    [
      "1.428724443",
      "0.701399341",
      "0.102895767"
    ],
    [
      "0.606997406",
      "0.622550817",
      "0.124854971"
    ],
    [
      "0.250730262",
      "0.506792605",
      "0.101173550"
    ],
    [
      "0.933871966",
      "0.860039085",
      "0.124854971"
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    [
      "0.496136183",
      "0.789852098",
      "0.075843632"
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    [
      "0.809016995",
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      "0.075843632"
    ],
    [
      "0.809016996",
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    ],
    [
      "0.809016995",
      "0.101173550",
      "0.101173550"
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    [
      "0.299899444",
      "1.041760052",
      "0.102895768"
    ],
    [
      "0.385570253",
      "0.654568074",
      "0.098874975"
    ],
    [
      "0.131567880",
      "0.908307520",
      "0.111918323"
    ],
    [
      "0.486204421",
      "0.964288195",
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    [
      "0.463976818",
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    [
      "0.331006568",
      "0.843505962",
      "0.097783920"
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    [
      "1.287027423",
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      "0.097783921"
    ],
    [
      "0.809016995",
      "0.475774994",
      "0.124854971"
    ],
    [
      "0.513590305",
      "0.281571002",
      "0.097783920"
    ],
    [
      "0.646187492",
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      "0.098874975"
    ],
    [
      "1.121897806",
      "0.789852098",
      "0.075843633"
    ],
    [
      "0.684162024",
      "0.860039085",
      "0.124854971"
    ],
    [
      "0.710142020",
      "0.275079133",
      "0.098874975"
    ],
    [
      "1.154057172",
      "1.163098019",
      "0.101173551"
    ],
    [
      "0.971846499",
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      "0.098874975"
    ],
    [
      "0.987955211",
      "1.281649381",
      "0.102895767"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 46
}

The current leader's answer

{
  "circles": [
    [
      "0.630078780",
      "1.281649381",
      "0.102895768"
    ],
    [
      "0.315426830",
      "0.313250256",
      "0.102895768"
    ],
    [
      "1.232463736",
      "0.654568074",
      "0.098874975"
    ],
    [
      "1.367303727",
      "0.506792603",
      "0.101173550"
    ],
    [
      "1.013079046",
      "0.102895767",
      "0.102895767"
    ],
    [
      "0.615646019",
      "0.422038644",
      "0.075843633"
    ],
    [
      "1.318134546",
      "1.041760052",
      "0.102895767"
    ],
    [
      "0.446678348",
      "0.466496696",
      "0.098874974"
    ],
    [
      "1.486466109",
      "0.908307520",
      "0.111918323"
    ],
    [
      "1.227703572",
      "0.111918324",
      "0.111918324"
    ],
    [
      "1.104443685",
      "0.281571003",
      "0.097783920"
    ],
    [
      "0.390330417",
      "0.111918324",
      "0.111918324"
    ],
    [
      "0.604954944",
      "0.102895767",
      "0.102895767"
    ],
    [
      "0.907891970",
      "0.275079133",
      "0.098874975"
    ],
    [
      "1.002387971",
      "0.422038644",
      "0.075843632"
    ],
    [
      "0.189309546",
      "0.701399343",
      "0.102895768"
    ],
    [
      "0.809016996",
      "1.400503111",
      "0.111918324"
    ],
    [
      "1.302607159",
      "0.313250255",
      "0.102895767"
    ],
    [
      "1.011036583",
      "0.622550816",
      "0.124854971"
    ],
    [
      "1.171355641",
      "0.466496695",
      "0.098874975"
    ],
    [
      "1.131829570",
      "0.964288195",
      "0.098874974"
    ],
    [
      "0.809016995",
      "0.688190959",
      "0.087560994"
    ],
    [
      "1.428724443",
      "0.701399341",
      "0.102895767"
    ],
    [
      "0.606997406",
      "0.622550817",
      "0.124854971"
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    [
      "0.250730262",
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      "0.098874975"
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    [
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      "1.080522699",
      "0.098874975"
    ],
    [
      "1.121897806",
      "0.789852098",
      "0.075843633"
    ],
    [
      "0.684162024",
      "0.860039085",
      "0.124854971"
    ],
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      "0.710142020",
      "0.275079133",
      "0.098874975"
    ],
    [
      "1.154057172",
      "1.163098019",
      "0.101173551"
    ],
    [
      "0.971846499",
      "1.080522698",
      "0.098874975"
    ],
    [
      "0.987955211",
      "1.281649381",
      "0.102895767"
    ]
  ]
}

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