P92 · Packing and covering · Classic · Hard

Sum of radii in the unit circle · n = 32

Place n non-overlapping circles inside a circle of radius 1, each with its own radius, making the sum of the radii as large as possible.

Instancen = 32

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	[5.191, 5.192]	See source attribution	Erich Friedman ↗
Site record	No player record yet	—	—

Formal definition

ContainerThe closed disc of radius 1 centred at (1, 1); a circle may touch the boundary
SubmissionExactly n circles, each [x, y, r] with r > 0, every number to at most nine decimals
ConstraintsEvery circle lies inside the disc, (x − 1)² + (y − 1)² ≤ (1 − r)², and no two interiors meet: the distance between centres is at least the sum of the radii. All checked exactly in integers; touching is allowed
ScoreThe sum of the radii, an exact integer sum shown to nine decimals; larger is better $S = Σ r$
ObjectiveMaximise the sum of the radii over every legal arrangement $\max_{S} Σ r$

Open the full editor ↗

Layout shown: starter layout

2y0

1234567891011121314151617181920212223242526272829303132

0x2

VERIFIED CONSTRUCTIONscored by the sum of the radii

Getting a feel for it

Where the room for improvement is

As in the square, equal circles are among the worst strategies: one large circle near the centre, or a ring of large ones, with small circles tucked into the gaps between them and the rim. A disc has no corners, so the big circle in a corner becomes a ring along the rim, and the trade of sizes plays out in a different geometry.

Where the frontier is

Erich Friedman's maximum-total-perimeter table runs to n = 50: n ≤ 4 and 6–10 are closed-form arrangements by Elser and Cantrell, n = 5 was improved by Haowei Lin in July 2026, and n = 33–50 were added by Jonathan Viquerat in September 2026. The table is still being rewritten and no row carries an optimality proof; the site exhibits the pictured arrangements, reconstructed and verified exactly.

Source

LAYOUT & HISTORY

Starter layout

Inspect the layout, solver notes and record history.

Challenge this record ↗Submit a proof / idea ↓Share a proof or idea in the discussion. Accepted contributions can earn proof points.

ANSWER FORMAT

How to write your answer

The container is the disc of radius 1 centred at (1, 1), so every coordinate lies in [0, 2] and none is negative. Coordinates and radii share one unit and are written as plain decimals, to at most nine places.

Submit circles: exactly n triples [x, y, r], every number a decimal string of at most nine places such as "0.25". The container is the closed disc of radius 1 centred at (1, 1).

Starter answer

{
  "circles": [
    [
      "0.999893251",
      "0.478199959",
      "0.210467962"
    ],
    [
      "1.811558224",
      "0.995971334",
      "0.188431776"
    ],
    [
      "0.188440225",
      "0.996303577",
      "0.188431806"
    ],
    [
      "1.721501689",
      "0.629508991",
      "0.188933865"
    ],
    [
      "0.278346754",
      "0.629804345",
      "0.188933892"
    ],
    [
      "1.598262061",
      "1.552965790",
      "0.185329110"
    ],
    [
      "0.401964457",
      "1.553210731",
      "0.185329131"
    ],
    [
      "0.706803838",
      "1.761587671",
      "0.183924163"
    ],
    [
      "1.293508067",
      "1.761467531",
      "0.183924153"
    ],
    [
      "0.706143328",
      "0.228903535",
      "0.174808202"
    ],
    [
      "1.293541114",
      "0.228783342",
      "0.174808192"
    ],
    [
      "1.369547276",
      "0.564127808",
      "0.169041843"
    ],
    [
      "0.630274415",
      "0.564279101",
      "0.169041855"
    ],
    [
      "0.527697805",
      "0.885739404",
      "0.168387673"
    ],
    [
      "1.472255454",
      "0.885546069",
      "0.168387657"
    ],
    [
      "0.722354862",
      "1.414021532",
      "0.163989697"
    ],
    [
      "1.277814716",
      "1.413907810",
      "0.163989688"
    ],
    [
      "0.466251504",
      "1.211122918",
      "0.162746859"
    ],
    [
      "1.533834955",
      "1.210904367",
      "0.162746842"
    ],
    [
      "1.000119076",
      "1.581408606",
      "0.160311691"
    ],
    [
      "0.843557434",
      "0.810094460",
      "0.156403741"
    ],
    [
      "1.156364905",
      "0.810030435",
      "0.156403736"
    ],
    [
      "0.758594876",
      "1.103286951",
      "0.148851026"
    ],
    [
      "1.241447480",
      "1.103188109",
      "0.148851019"
    ],
    [
      "1.000056169",
      "1.274148595",
      "0.146948325"
    ],
    [
      "0.999822792",
      "0.133866010",
      "0.133865991"
    ],
    [
      "1.808810755",
      "1.316030668",
      "0.131639348"
    ],
    [
      "0.191318735",
      "1.316361810",
      "0.131639370"
    ],
    [
      "1.000178346",
      "1.870860136",
      "0.129139845"
    ],
    [
      "1.574555284",
      "0.346566691",
      "0.129891464"
    ],
    [
      "0.425177362",
      "0.346801889",
      "0.129891479"
    ],
    [
      "1.000004348",
      "1.021004313",
      "0.106195960"
    ]
  ]
}

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

Instance parameters

{
  "n": 32
}

Starter answer

{
  "circles": [
    [
      "0.999893251",
      "0.478199959",
      "0.210467962"
    ],
    [
      "1.811558224",
      "0.995971334",
      "0.188431776"
    ],
    [
      "0.188440225",
      "0.996303577",
      "0.188431806"
    ],
    [
      "1.721501689",
      "0.629508991",
      "0.188933865"
    ],
    [
      "0.278346754",
      "0.629804345",
      "0.188933892"
    ],
    [
      "1.598262061",
      "1.552965790",
      "0.185329110"
    ],
    [
      "0.401964457",
      "1.553210731",
      "0.185329131"
    ],
    [
      "0.706803838",
      "1.761587671",
      "0.183924163"
    ],
    [
      "1.293508067",
      "1.761467531",
      "0.183924153"
    ],
    [
      "0.706143328",
      "0.228903535",
      "0.174808202"
    ],
    [
      "1.293541114",
      "0.228783342",
      "0.174808192"
    ],
    [
      "1.369547276",
      "0.564127808",
      "0.169041843"
    ],
    [
      "0.630274415",
      "0.564279101",
      "0.169041855"
    ],
    [
      "0.527697805",
      "0.885739404",
      "0.168387673"
    ],
    [
      "1.472255454",
      "0.885546069",
      "0.168387657"
    ],
    [
      "0.722354862",
      "1.414021532",
      "0.163989697"
    ],
    [
      "1.277814716",
      "1.413907810",
      "0.163989688"
    ],
    [
      "0.466251504",
      "1.211122918",
      "0.162746859"
    ],
    [
      "1.533834955",
      "1.210904367",
      "0.162746842"
    ],
    [
      "1.000119076",
      "1.581408606",
      "0.160311691"
    ],
    [
      "0.843557434",
      "0.810094460",
      "0.156403741"
    ],
    [
      "1.156364905",
      "0.810030435",
      "0.156403736"
    ],
    [
      "0.758594876",
      "1.103286951",
      "0.148851026"
    ],
    [
      "1.241447480",
      "1.103188109",
      "0.148851019"
    ],
    [
      "1.000056169",
      "1.274148595",
      "0.146948325"
    ],
    [
      "0.999822792",
      "0.133866010",
      "0.133865991"
    ],
    [
      "1.808810755",
      "1.316030668",
      "0.131639348"
    ],
    [
      "0.191318735",
      "1.316361810",
      "0.131639370"
    ],
    [
      "1.000178346",
      "1.870860136",
      "0.129139845"
    ],
    [
      "1.574555284",
      "0.346566691",
      "0.129891464"
    ],
    [
      "0.425177362",
      "0.346801889",
      "0.129891479"
    ],
    [
      "1.000004348",
      "1.021004313",
      "0.106195960"
    ]
  ]
}

Submit circles: exactly n triples [x, y, r], every number a decimal string of at most nine places such as "0.25". The container is the closed disc of radius 1 centred at (1, 1). · Verifier v1.0.0

DISCUSSION

Discussion

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对本周挑战的评价NUE_13#311 replies1h ago