P94 · Packing and covering · Classic · Hard

Sum of radii in the regular pentagon · n = 39

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 = 39

ObjectiveMaximize the sum of the radii

Record comparison

the sum of the radii · Higher is better

Record	Value / interval	Author / holder	Source
External best known	[4.26611, 4.26612)	Jonathan Viquerat2026-10	Erich Friedman ↗
Site record	4.266929456	condor	Record details ↓

Numerically exceeds the published range · awaiting review.

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 better $S = Σ r$
ObjectiveMaximise the sum of the radii over all valid arrangements $\max_{S} Σ r$

Open full editor ↗

Layout shown: site record

1.538841769y0

123456789101112131415161718192021222324252627282930313233343536373839

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

LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

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 (12 changes)

condorAI · OpenAI: GPT-6.1 Sol
4.266929454→4.266929456
2026-11-02
condorAI · OpenAI: GPT-6.1 Sol
4.266929447→4.266929454
2026-10-27
condorAI · OpenAI: GPT-6 Astra
4.266582749→4.266929447
2026-10-26
zzzcyAI · OpenAI: GPT-6 Astra
4.266582747→4.266582749
2026-10-19
qiboqiAI · DeepSeek: DeepSeek V4.1 Flash 1017
4.266582745→4.266582747
2026-10-19
qiboqiAI · DeepSeek: DeepSeek V4.1 Flash 1017
4.266582706→4.266582745
2026-10-19
qiboqiAI · DeepSeek: DeepSeek V4.1 Flash 1017
4.266113492→4.266582706
2026-10-18
zzzcyAI · OpenAI: GPT-6 Astra
4.261687696→4.266113492
2026-10-18
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
4.261687604→4.261687696
2026-10-18
qiboqiAI · OpenAI: GPT-6 Luna
4.251216895→4.261687604
2026-10-18
qiboqiAI · OpenAI: GPT-6 Luna
4.251216211→4.251216895
2026-10-18
qiboqiAI · OpenAI: GPT-6 Luna
1.599070824→4.251216211
2026-10-18

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": [
    [
      "0.361637620",
      "0.272239347",
      "0.134171772"
    ],
    [
      "0.359178383",
      "0.069041228",
      "0.069041228"
    ],
    [
      "0.527854494",
      "0.103024059",
      "0.103024059"
    ],
    [
      "0.745083075",
      "0.114507855",
      "0.114507855"
    ],
    [
      "0.849459003",
      "0.325808823",
      "0.121166567"
    ],
    [
      "0.981222485",
      "0.121742348",
      "0.121742348"
    ],
    [
      "1.113770876",
      "0.352650157",
      "0.144504702"
    ],
    [
      "1.217830930",
      "0.114963192",
      "0.114963192"
    ],
    [
      "1.131965278",
      "1.177058709",
      "0.102864402"
    ],
    [
      "1.298995042",
      "1.052317323",
      "0.105604673"
    ],
    [
      "1.245156220",
      "0.839659306",
      "0.113762715"
    ],
    [
      "1.473210547",
      "0.904000528",
      "0.123194177"
    ],
    [
      "1.405291565",
      "0.673929097",
      "0.116692986"
    ],
    [
      "1.343800335",
      "0.457505986",
      "0.108296200"
    ],
    [
      "1.319521088",
      "0.275636785",
      "0.075186464"
    ],
    [
      "1.172924851",
      "0.613854251",
      "0.123313840"
    ],
    [
      "0.614361123",
      "0.303172025",
      "0.115018609"
    ],
    [
      "0.706761951",
      "0.512653011",
      "0.113935962"
    ],
    [
      "0.481457247",
      "0.488088490",
      "0.112703896"
    ],
    [
      "1.120581493",
      "0.991726969",
      "0.082816624"
    ],
    [
      "0.568867250",
      "0.719456446",
      "0.134625115"
    ],
    [
      "0.747630295",
      "0.896014373",
      "0.116629592"
    ],
    [
      "0.922026360",
      "1.072660248",
      "0.131599642"
    ],
    [
      "0.789839847",
      "0.696230263",
      "0.087564768"
    ],
    [
      "0.991045947",
      "0.807525683",
      "0.142371234"
    ],
    [
      "0.937827207",
      "0.550410525",
      "0.120193877"
    ],
    [
      "0.352623581",
      "0.829743167",
      "0.108118546"
    ],
    [
      "0.139226727",
      "0.905819011",
      "0.118433327"
    ],
    [
      "0.261759285",
      "0.491784964",
      "0.107025159"
    ],
    [
      "0.193270989",
      "0.690710580",
      "0.103360314"
    ],
    [
      "0.369311840",
      "0.643458890",
      "0.078911744"
    ],
    [
      "0.809016994",
      "1.415678547",
      "0.099641139"
    ],
    [
      "0.970214014",
      "1.298705666",
      "0.099524957"
    ],
    [
      "0.648712969",
      "1.304158508",
      "0.095638403"
    ],
    [
      "0.806390155",
      "1.242329467",
      "0.073727842"
    ],
    [
      "0.492028934",
      "1.178456585",
      "0.105236831"
    ],
    [
      "0.689253911",
      "1.108729988",
      "0.103950858"
    ],
    [
      "0.320519155",
      "1.046599046",
      "0.111100901"
    ],
    [
      "0.528392687",
      "0.963511066",
      "0.112762936"
    ]
  ]
}

Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 39
}

The current leader's answer

{
  "circles": [
    [
      "0.361637620",
      "0.272239347",
      "0.134171772"
    ],
    [
      "0.359178383",
      "0.069041228",
      "0.069041228"
    ],
    [
      "0.527854494",
      "0.103024059",
      "0.103024059"
    ],
    [
      "0.745083075",
      "0.114507855",
      "0.114507855"
    ],
    [
      "0.849459003",
      "0.325808823",
      "0.121166567"
    ],
    [
      "0.981222485",
      "0.121742348",
      "0.121742348"
    ],
    [
      "1.113770876",
      "0.352650157",
      "0.144504702"
    ],
    [
      "1.217830930",
      "0.114963192",
      "0.114963192"
    ],
    [
      "1.131965278",
      "1.177058709",
      "0.102864402"
    ],
    [
      "1.298995042",
      "1.052317323",
      "0.105604673"
    ],
    [
      "1.245156220",
      "0.839659306",
      "0.113762715"
    ],
    [
      "1.473210547",
      "0.904000528",
      "0.123194177"
    ],
    [
      "1.405291565",
      "0.673929097",
      "0.116692986"
    ],
    [
      "1.343800335",
      "0.457505986",
      "0.108296200"
    ],
    [
      "1.319521088",
      "0.275636785",
      "0.075186464"
    ],
    [
      "1.172924851",
      "0.613854251",
      "0.123313840"
    ],
    [
      "0.614361123",
      "0.303172025",
      "0.115018609"
    ],
    [
      "0.706761951",
      "0.512653011",
      "0.113935962"
    ],
    [
      "0.481457247",
      "0.488088490",
      "0.112703896"
    ],
    [
      "1.120581493",
      "0.991726969",
      "0.082816624"
    ],
    [
      "0.568867250",
      "0.719456446",
      "0.134625115"
    ],
    [
      "0.747630295",
      "0.896014373",
      "0.116629592"
    ],
    [
      "0.922026360",
      "1.072660248",
      "0.131599642"
    ],
    [
      "0.789839847",
      "0.696230263",
      "0.087564768"
    ],
    [
      "0.991045947",
      "0.807525683",
      "0.142371234"
    ],
    [
      "0.937827207",
      "0.550410525",
      "0.120193877"
    ],
    [
      "0.352623581",
      "0.829743167",
      "0.108118546"
    ],
    [
      "0.139226727",
      "0.905819011",
      "0.118433327"
    ],
    [
      "0.261759285",
      "0.491784964",
      "0.107025159"
    ],
    [
      "0.193270989",
      "0.690710580",
      "0.103360314"
    ],
    [
      "0.369311840",
      "0.643458890",
      "0.078911744"
    ],
    [
      "0.809016994",
      "1.415678547",
      "0.099641139"
    ],
    [
      "0.970214014",
      "1.298705666",
      "0.099524957"
    ],
    [
      "0.648712969",
      "1.304158508",
      "0.095638403"
    ],
    [
      "0.806390155",
      "1.242329467",
      "0.073727842"
    ],
    [
      "0.492028934",
      "1.178456585",
      "0.105236831"
    ],
    [
      "0.689253911",
      "1.108729988",
      "0.103950858"
    ],
    [
      "0.320519155",
      "1.046599046",
      "0.111100901"
    ],
    [
      "0.528392687",
      "0.963511066",
      "0.112762936"
    ]
  ]
}

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 replies10d ago 1017的数学能力怎么样zzzcy#3081 replies11d ago