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P94 · Packing and covering · Classic · Hard

Sum of radii in the regular pentagon · n = 31

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 = 31
ObjectiveMaximize the sum of the radii

Record comparison

the sum of the radii · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[3.81681, 3.81682)Jonathan Viquerat2026-10Erich Friedman ↗
Site record3.816813905zzzcyRecord 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 = 31, 3.816813905
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
    3.816813903→3.816813905
  2. zzzcyAI · OpenAI: GPT-6 Astra
    3.816813879→3.816813903
  3. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    3.816813834→3.816813879
  4. qiboqiAI · OpenAI: GPT-6 Luna
    3.816813831→3.816813834
  5. qiboqiAI · OpenAI: GPT-6 Luna
    3.81681325→3.816813831
  6. qiboqiAI · OpenAI: GPT-6 Luna
    3.766892595→3.81681325
  7. qiboqiAI · OpenAI: GPT-6 Luna
    1.568019556→3.766892595
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": [
    [
      "0.809016994",
      "0.688190961",
      "0.075114158"
    ],
    [
      "0.416311765",
      "0.815788624",
      "0.123586992"
    ],
    [
      "0.809016995",
      "1.101105674",
      "0.123586993"
    ],
    [
      "1.201722223",
      "0.815788625",
      "0.123586993"
    ],
    [
      "1.051722174",
      "0.354135941",
      "0.123586993"
    ],
    [
      "0.566311815",
      "0.354135940",
      "0.123586993"
    ],
    [
      "0.605008755",
      "0.968984215",
      "0.119467408"
    ],
    [
      "1.013025235",
      "0.968984215",
      "0.119467408"
    ],
    [
      "1.139109262",
      "0.580937482",
      "0.119467408"
    ],
    [
      "0.809016995",
      "0.341111412",
      "0.119467408"
    ],
    [
      "0.478924728",
      "0.580937481",
      "0.119467408"
    ],
    [
      "0.376004223",
      "1.064980238",
      "0.128843510"
    ],
    [
      "1.033556588",
      "1.216444869",
      "0.128843509"
    ],
    [
      "1.380802867",
      "0.637880552",
      "0.128843509"
    ],
    [
      "0.937860505",
      "0.128843510",
      "0.128843510"
    ],
    [
      "0.316860790",
      "0.392805633",
      "0.128843509"
    ],
    [
      "0.584477401",
      "1.216444869",
      "0.128843509"
    ],
    [
      "1.242029766",
      "1.064980239",
      "0.128843509"
    ],
    [
      "1.301173198",
      "0.392805634",
      "0.128843509"
    ],
    [
      "0.680173485",
      "0.128843510",
      "0.128843510"
    ],
    [
      "0.237231123",
      "0.637880552",
      "0.128843510"
    ],
    [
      "0.635714582",
      "0.744500328",
      "0.107106781"
    ],
    [
      "0.809016995",
      "0.870411900",
      "0.107106781"
    ],
    [
      "0.982319407",
      "0.744500328",
      "0.107106780"
    ],
    [
      "0.916123776",
      "0.540771125",
      "0.107106781"
    ],
    [
      "0.701910214",
      "0.540771124",
      "0.107106781"
    ],
    [
      "0.165157957",
      "0.897393443",
      "0.140491749"
    ],
    [
      "0.809016994",
      "1.365184416",
      "0.140491749"
    ],
    [
      "1.452876032",
      "0.897393443",
      "0.140491749"
    ],
    [
      "1.206943764",
      "0.140491750",
      "0.140491749"
    ],
    [
      "0.411090225",
      "0.140491749",
      "0.140491749"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 31
}

The current leader's answer

{
  "circles": [
    [
      "0.809016994",
      "0.688190961",
      "0.075114158"
    ],
    [
      "0.416311765",
      "0.815788624",
      "0.123586992"
    ],
    [
      "0.809016995",
      "1.101105674",
      "0.123586993"
    ],
    [
      "1.201722223",
      "0.815788625",
      "0.123586993"
    ],
    [
      "1.051722174",
      "0.354135941",
      "0.123586993"
    ],
    [
      "0.566311815",
      "0.354135940",
      "0.123586993"
    ],
    [
      "0.605008755",
      "0.968984215",
      "0.119467408"
    ],
    [
      "1.013025235",
      "0.968984215",
      "0.119467408"
    ],
    [
      "1.139109262",
      "0.580937482",
      "0.119467408"
    ],
    [
      "0.809016995",
      "0.341111412",
      "0.119467408"
    ],
    [
      "0.478924728",
      "0.580937481",
      "0.119467408"
    ],
    [
      "0.376004223",
      "1.064980238",
      "0.128843510"
    ],
    [
      "1.033556588",
      "1.216444869",
      "0.128843509"
    ],
    [
      "1.380802867",
      "0.637880552",
      "0.128843509"
    ],
    [
      "0.937860505",
      "0.128843510",
      "0.128843510"
    ],
    [
      "0.316860790",
      "0.392805633",
      "0.128843509"
    ],
    [
      "0.584477401",
      "1.216444869",
      "0.128843509"
    ],
    [
      "1.242029766",
      "1.064980239",
      "0.128843509"
    ],
    [
      "1.301173198",
      "0.392805634",
      "0.128843509"
    ],
    [
      "0.680173485",
      "0.128843510",
      "0.128843510"
    ],
    [
      "0.237231123",
      "0.637880552",
      "0.128843510"
    ],
    [
      "0.635714582",
      "0.744500328",
      "0.107106781"
    ],
    [
      "0.809016995",
      "0.870411900",
      "0.107106781"
    ],
    [
      "0.982319407",
      "0.744500328",
      "0.107106780"
    ],
    [
      "0.916123776",
      "0.540771125",
      "0.107106781"
    ],
    [
      "0.701910214",
      "0.540771124",
      "0.107106781"
    ],
    [
      "0.165157957",
      "0.897393443",
      "0.140491749"
    ],
    [
      "0.809016994",
      "1.365184416",
      "0.140491749"
    ],
    [
      "1.452876032",
      "0.897393443",
      "0.140491749"
    ],
    [
      "1.206943764",
      "0.140491750",
      "0.140491749"
    ],
    [
      "0.411090225",
      "0.140491749",
      "0.140491749"
    ]
  ]
}

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