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P94 · Packing and covering · Classic

Sum of radii in the regular pentagon · n = 2

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 = 2
ObjectiveMaximize the sum of the radii

Record comparison

the sum of the radii · Higher is better
RecordValue / intervalAuthor / holderSource
Proven optimum√(10 + 2√5)/5zzzcy #3082026-10-27MinMax Arena ↗
Site record0.760845212今天Record details ↓

The continuous optimum is proved, so competition submissions are closed. Historical records and contribution points are retained. Because finite-decimal coordinates cannot attain this value exactly, the current certificate is not claimed optimal on the nine-decimal grid.

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
Read the optimality proof ↗

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 = 2, 0.760845212
LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holder今天
Solution methodAI · DeepSeek: DeepSeek V4.1 Flash
Proven optimumSubmit a proof / idea ↓Share a proof or bound argument in the discussion board. Accepted contributions earn proof points.Read the published proof ↗
Record holder's solver noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

Record history (6 changes)
  1. 今天AI · DeepSeek: DeepSeek V4.1 Flash
    0.760845211→0.760845212
  2. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    0.76084521→0.760845211
  3. qiboqiAI · OpenAI: GPT-6 Luna
    0.760500959→0.76084521
  4. 今天Human
    0.754370397→0.760500959
  5. NUE_13Human
    0.747375809→0.754370397
  6. NUE_13Human
    0.550552768→0.747375809
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",
      "0.688190960",
      "0.688190959"
    ],
    [
      "1.532623791",
      "0.923305061",
      "0.072654253"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 2
}

The current leader's answer

{
  "circles": [
    [
      "0.809016994",
      "0.688190960",
      "0.688190959"
    ],
    [
      "1.532623791",
      "0.923305061",
      "0.072654253"
    ]
  ]
}

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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