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P94 · Packing and covering · Classic · Hard

Sum of radii in the regular pentagon · n = 27

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 = 27
ObjectiveMaximize the sum of the radii

Record comparison

the sum of the radii · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[3.53987, 3.53988)Jonathan Viquerat2026-10Erich Friedman ↗
Site record3.539868661zzzcyRecord 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 = 27, 3.539868661
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 (6 changes)
  1. zzzcyAI · OpenAI: GPT-6 Astra
    3.53986866→3.539868661
  2. zzzcyAI · OpenAI: GPT-6 Astra
    3.539868659→3.53986866
  3. zzzcyAI · OpenAI: GPT-6 Astra
    3.53956838→3.539868659
  4. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    3.539568332→3.53956838
  5. qiboqiAI · OpenAI: GPT-6 Luna
    3.539567858→3.539568332
  6. qiboqiAI · OpenAI: GPT-6 Luna
    1.546208568→3.539567858
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": [
    [
      "0.289061763",
      "0.609539722",
      "0.169379580"
    ],
    [
      "1.328972225",
      "0.609539721",
      "0.169379580"
    ],
    [
      "0.553225750",
      "0.404073524",
      "0.165282892"
    ],
    [
      "1.064808237",
      "0.404073523",
      "0.165282893"
    ],
    [
      "0.167438151",
      "0.896652563",
      "0.142431398"
    ],
    [
      "1.450595838",
      "0.896652563",
      "0.142431398"
    ],
    [
      "0.409029262",
      "0.137655079",
      "0.137655077"
    ],
    [
      "1.209004725",
      "0.137655078",
      "0.137655078"
    ],
    [
      "0.577798780",
      "0.702209627",
      "0.133864177"
    ],
    [
      "1.040235207",
      "0.702209627",
      "0.133864178"
    ],
    [
      "0.677811976",
      "0.131205017",
      "0.131205017"
    ],
    [
      "0.940222010",
      "0.131205017",
      "0.131205017"
    ],
    [
      "0.580137864",
      "1.211615033",
      "0.130200212"
    ],
    [
      "0.809016994",
      "1.088402090",
      "0.129736483"
    ],
    [
      "0.809016993",
      "0.828885077",
      "0.129780530"
    ],
    [
      "1.037896123",
      "1.211615035",
      "0.130200212"
    ],
    [
      "0.371006862",
      "1.067870247",
      "0.123568067"
    ],
    [
      "1.247027126",
      "1.067870248",
      "0.123568068"
    ],
    [
      "0.592095205",
      "0.958686486",
      "0.123010822"
    ],
    [
      "1.025938783",
      "0.958686485",
      "0.123010823"
    ],
    [
      "0.809016994",
      "1.361561402",
      "0.143422829"
    ],
    [
      "0.809016993",
      "0.563160328",
      "0.135944217"
    ],
    [
      "0.401558221",
      "0.850178842",
      "0.096256711"
    ],
    [
      "0.298577253",
      "0.343967530",
      "0.096363028"
    ],
    [
      "1.216475769",
      "0.850178842",
      "0.096256711"
    ],
    [
      "1.319456735",
      "0.343967529",
      "0.096363028"
    ],
    [
      "0.809016993",
      "0.324665474",
      "0.102550635"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 27
}

The current leader's answer

{
  "circles": [
    [
      "0.289061763",
      "0.609539722",
      "0.169379580"
    ],
    [
      "1.328972225",
      "0.609539721",
      "0.169379580"
    ],
    [
      "0.553225750",
      "0.404073524",
      "0.165282892"
    ],
    [
      "1.064808237",
      "0.404073523",
      "0.165282893"
    ],
    [
      "0.167438151",
      "0.896652563",
      "0.142431398"
    ],
    [
      "1.450595838",
      "0.896652563",
      "0.142431398"
    ],
    [
      "0.409029262",
      "0.137655079",
      "0.137655077"
    ],
    [
      "1.209004725",
      "0.137655078",
      "0.137655078"
    ],
    [
      "0.577798780",
      "0.702209627",
      "0.133864177"
    ],
    [
      "1.040235207",
      "0.702209627",
      "0.133864178"
    ],
    [
      "0.677811976",
      "0.131205017",
      "0.131205017"
    ],
    [
      "0.940222010",
      "0.131205017",
      "0.131205017"
    ],
    [
      "0.580137864",
      "1.211615033",
      "0.130200212"
    ],
    [
      "0.809016994",
      "1.088402090",
      "0.129736483"
    ],
    [
      "0.809016993",
      "0.828885077",
      "0.129780530"
    ],
    [
      "1.037896123",
      "1.211615035",
      "0.130200212"
    ],
    [
      "0.371006862",
      "1.067870247",
      "0.123568067"
    ],
    [
      "1.247027126",
      "1.067870248",
      "0.123568068"
    ],
    [
      "0.592095205",
      "0.958686486",
      "0.123010822"
    ],
    [
      "1.025938783",
      "0.958686485",
      "0.123010823"
    ],
    [
      "0.809016994",
      "1.361561402",
      "0.143422829"
    ],
    [
      "0.809016993",
      "0.563160328",
      "0.135944217"
    ],
    [
      "0.401558221",
      "0.850178842",
      "0.096256711"
    ],
    [
      "0.298577253",
      "0.343967530",
      "0.096363028"
    ],
    [
      "1.216475769",
      "0.850178842",
      "0.096256711"
    ],
    [
      "1.319456735",
      "0.343967529",
      "0.096363028"
    ],
    [
      "0.809016993",
      "0.324665474",
      "0.102550635"
    ]
  ]
}

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