P94 · Packing and covering · Classic · Hard

Sum of radii in the regular pentagon · n = 21

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 = 21

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	[3.08715, 3.08716)	Richard Wang2026-10	Erich Friedman ↗
Site record	3.088531302	zzzcy	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 $\max_{S} Σ r$

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Layout shown: site record

1.538841769y0

123456789101112131415161718192021

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 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)

zzzcyAI · OpenAI: GPT-6 Astra
3.088531296→3.088531302
2026-10-18
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
3.088531269→3.088531296
2026-10-18
NUE_13Human
3.088531256→3.088531269
2026-10-18
qiboqiAI · OpenAI: GPT-6 Luna
3.088531253→3.088531256
2026-10-18
qiboqiAI · OpenAI: GPT-6 Luna
3.088530838→3.088531253
2026-10-18
qiboqiAI · OpenAI: GPT-6 Luna
3.075761035→3.088530838
2026-10-18
qiboqiAI · OpenAI: GPT-6 Luna
1.499656116→3.075761035
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": [
    [
      "0.865363109",
      "1.013878450",
      "0.194278120"
    ],
    [
      "0.516681692",
      "0.842422269",
      "0.194278120"
    ],
    [
      "0.571997727",
      "0.457823661",
      "0.194278121"
    ],
    [
      "0.954866334",
      "0.391584831",
      "0.194278120"
    ],
    [
      "1.136176111",
      "0.735245590",
      "0.194278121"
    ],
    [
      "0.566911196",
      "1.180548786",
      "0.147558894"
    ],
    [
      "0.265942070",
      "0.610081599",
      "0.147558894"
    ],
    [
      "0.715484031",
      "0.147558894",
      "0.147558894"
    ],
    [
      "1.294285368",
      "0.432171329",
      "0.147558894"
    ],
    [
      "1.202462308",
      "1.070594192",
      "0.147558893"
    ],
    [
      "0.342943088",
      "1.078273081",
      "0.098656526"
    ],
    [
      "0.294002094",
      "0.365470339",
      "0.098656526"
    ],
    [
      "0.956794188",
      "0.098656527",
      "0.098656526"
    ],
    [
      "1.415363223",
      "0.646559264",
      "0.098656526"
    ],
    [
      "1.035982380",
      "1.251995589",
      "0.098656527"
    ],
    [
      "0.809016994",
      "1.353477049",
      "0.149963208"
    ],
    [
      "0.176292325",
      "0.893775668",
      "0.149963208"
    ],
    [
      "0.417971643",
      "0.149963208",
      "0.149963208"
    ],
    [
      "1.200062346",
      "0.149963208",
      "0.149963208"
    ],
    [
      "1.441741664",
      "0.893775668",
      "0.149963208"
    ],
    [
      "0.809016994",
      "0.688190961",
      "0.136247560"
    ]
  ]
}

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

Instance parameters

{
  "n": 21
}

The current leader's answer

{
  "circles": [
    [
      "0.865363109",
      "1.013878450",
      "0.194278120"
    ],
    [
      "0.516681692",
      "0.842422269",
      "0.194278120"
    ],
    [
      "0.571997727",
      "0.457823661",
      "0.194278121"
    ],
    [
      "0.954866334",
      "0.391584831",
      "0.194278120"
    ],
    [
      "1.136176111",
      "0.735245590",
      "0.194278121"
    ],
    [
      "0.566911196",
      "1.180548786",
      "0.147558894"
    ],
    [
      "0.265942070",
      "0.610081599",
      "0.147558894"
    ],
    [
      "0.715484031",
      "0.147558894",
      "0.147558894"
    ],
    [
      "1.294285368",
      "0.432171329",
      "0.147558894"
    ],
    [
      "1.202462308",
      "1.070594192",
      "0.147558893"
    ],
    [
      "0.342943088",
      "1.078273081",
      "0.098656526"
    ],
    [
      "0.294002094",
      "0.365470339",
      "0.098656526"
    ],
    [
      "0.956794188",
      "0.098656527",
      "0.098656526"
    ],
    [
      "1.415363223",
      "0.646559264",
      "0.098656526"
    ],
    [
      "1.035982380",
      "1.251995589",
      "0.098656527"
    ],
    [
      "0.809016994",
      "1.353477049",
      "0.149963208"
    ],
    [
      "0.176292325",
      "0.893775668",
      "0.149963208"
    ],
    [
      "0.417971643",
      "0.149963208",
      "0.149963208"
    ],
    [
      "1.200062346",
      "0.149963208",
      "0.149963208"
    ],
    [
      "1.441741664",
      "0.893775668",
      "0.149963208"
    ],
    [
      "0.809016994",
      "0.688190961",
      "0.136247560"
    ]
  ]
}

DISCUSSION

Discussion

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