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P94 · Packing and covering · Classic · Hard

Sum of radii in the regular pentagon · n = 30

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 = 30
ObjectiveMaximize the sum of the radii

Record comparison

the sum of the radii · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[3.74588, 3.74589)Jonathan Viquerat2026-10Erich Friedman ↗
Site record3.745880288zzzcyRecord 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 = 30, 3.745880288
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
    3.745880287→3.745880288
  2. qiboqiAI · DeepSeek: DeepSeek V4.1 Flash 1017
    3.745880286→3.745880287
  3. zzzcyAI · OpenAI: GPT-6 Astra
    3.74169973→3.745880286
  4. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    3.741699672→3.74169973
  5. qiboqiAI · OpenAI: GPT-6 Luna
    3.74169914→3.741699672
  6. qiboqiAI · OpenAI: GPT-6 Luna
    3.718583805→3.74169914
  7. qiboqiAI · OpenAI: GPT-6 Luna
    1.56306795→3.718583805
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": [
    [
      "1.052873730",
      "0.362446060",
      "0.123970183"
    ],
    [
      "0.981763727",
      "0.737716533",
      "0.110644640"
    ],
    [
      "0.802736938",
      "0.867787108",
      "0.110644640"
    ],
    [
      "0.628914905",
      "0.777095186",
      "0.085414340"
    ],
    [
      "0.777675581",
      "0.645053204",
      "0.113494744"
    ],
    [
      "0.949224546",
      "0.544376611",
      "0.085414340"
    ],
    [
      "1.033127649",
      "1.216376932",
      "0.129150595"
    ],
    [
      "0.378666226",
      "1.067616171",
      "0.128275681"
    ],
    [
      "0.321154835",
      "0.376254982",
      "0.127812956"
    ],
    [
      "1.382558424",
      "0.639316853",
      "0.127617716"
    ],
    [
      "1.011781974",
      "0.967273012",
      "0.120866206"
    ],
    [
      "0.604545622",
      "0.976364488",
      "0.115339530"
    ],
    [
      "0.526129213",
      "0.564650911",
      "0.150588771"
    ],
    [
      "0.778940559",
      "0.380972718",
      "0.150588771"
    ],
    [
      "1.453313611",
      "0.897535621",
      "0.140119522"
    ],
    [
      "0.809016993",
      "1.365644512",
      "0.140119523"
    ],
    [
      "0.168168463",
      "0.896415270",
      "0.143052638"
    ],
    [
      "0.404244312",
      "0.131069158",
      "0.131069158"
    ],
    [
      "1.205083169",
      "0.143052638",
      "0.143052638"
    ],
    [
      "1.242097704",
      "1.064551300",
      "0.129150595"
    ],
    [
      "0.585300905",
      "1.218558344",
      "0.127617715"
    ],
    [
      "0.247238455",
      "0.629150414",
      "0.135663287"
    ],
    [
      "1.302857515",
      "0.396151896",
      "0.128275681"
    ],
    [
      "0.538887709",
      "0.316389893",
      "0.097999869"
    ],
    [
      "0.807789154",
      "1.101952550",
      "0.123575297"
    ],
    [
      "1.202907076",
      "0.814882577",
      "0.123575297"
    ],
    [
      "0.423859310",
      "0.819451787",
      "0.123970183"
    ],
    [
      "0.926465214",
      "0.135663287",
      "0.135663287"
    ],
    [
      "0.663105948",
      "0.127812956",
      "0.127812956"
    ],
    [
      "1.146271435",
      "0.582777646",
      "0.115339529"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 30
}

The current leader's answer

{
  "circles": [
    [
      "1.052873730",
      "0.362446060",
      "0.123970183"
    ],
    [
      "0.981763727",
      "0.737716533",
      "0.110644640"
    ],
    [
      "0.802736938",
      "0.867787108",
      "0.110644640"
    ],
    [
      "0.628914905",
      "0.777095186",
      "0.085414340"
    ],
    [
      "0.777675581",
      "0.645053204",
      "0.113494744"
    ],
    [
      "0.949224546",
      "0.544376611",
      "0.085414340"
    ],
    [
      "1.033127649",
      "1.216376932",
      "0.129150595"
    ],
    [
      "0.378666226",
      "1.067616171",
      "0.128275681"
    ],
    [
      "0.321154835",
      "0.376254982",
      "0.127812956"
    ],
    [
      "1.382558424",
      "0.639316853",
      "0.127617716"
    ],
    [
      "1.011781974",
      "0.967273012",
      "0.120866206"
    ],
    [
      "0.604545622",
      "0.976364488",
      "0.115339530"
    ],
    [
      "0.526129213",
      "0.564650911",
      "0.150588771"
    ],
    [
      "0.778940559",
      "0.380972718",
      "0.150588771"
    ],
    [
      "1.453313611",
      "0.897535621",
      "0.140119522"
    ],
    [
      "0.809016993",
      "1.365644512",
      "0.140119523"
    ],
    [
      "0.168168463",
      "0.896415270",
      "0.143052638"
    ],
    [
      "0.404244312",
      "0.131069158",
      "0.131069158"
    ],
    [
      "1.205083169",
      "0.143052638",
      "0.143052638"
    ],
    [
      "1.242097704",
      "1.064551300",
      "0.129150595"
    ],
    [
      "0.585300905",
      "1.218558344",
      "0.127617715"
    ],
    [
      "0.247238455",
      "0.629150414",
      "0.135663287"
    ],
    [
      "1.302857515",
      "0.396151896",
      "0.128275681"
    ],
    [
      "0.538887709",
      "0.316389893",
      "0.097999869"
    ],
    [
      "0.807789154",
      "1.101952550",
      "0.123575297"
    ],
    [
      "1.202907076",
      "0.814882577",
      "0.123575297"
    ],
    [
      "0.423859310",
      "0.819451787",
      "0.123970183"
    ],
    [
      "0.926465214",
      "0.135663287",
      "0.135663287"
    ],
    [
      "0.663105948",
      "0.127812956",
      "0.127812956"
    ],
    [
      "1.146271435",
      "0.582777646",
      "0.115339529"
    ]
  ]
}

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