P92 · Packing and covering · Classic · Hard

Sum of radii in the unit circle · n = 34

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 = 34

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	[5.360435, 5.360445]	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

12345678910111213141516171819202122232425262728293031323334

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": [
    [
      "1.255654784",
      "0.920195108",
      "0.201433794"
    ],
    [
      "1.582741185",
      "1.572355598",
      "0.183190218"
    ],
    [
      "1.153652641",
      "0.197772459",
      "0.183190253"
    ],
    [
      "0.579922085",
      "0.296174502",
      "0.180344104"
    ],
    [
      "1.054915799",
      "1.817814223",
      "0.180344066"
    ],
    [
      "1.777492444",
      "1.269022454",
      "0.177280374"
    ],
    [
      "0.701655130",
      "1.767589400",
      "0.176469096"
    ],
    [
      "1.486380982",
      "0.336448131",
      "0.177280397"
    ],
    [
      "0.317966369",
      "0.538444776",
      "0.176469126"
    ],
    [
      "1.428103928",
      "1.252014525",
      "0.172521861"
    ],
    [
      "0.175823729",
      "1.076826171",
      "0.172250770"
    ],
    [
      "1.208713194",
      "0.549197354",
      "0.172521878"
    ],
    [
      "0.278477817",
      "1.405677877",
      "0.172250762"
    ],
    [
      "0.915536531",
      "1.504377065",
      "0.162685716"
    ],
    [
      "0.643603605",
      "0.633241464",
      "0.162685736"
    ],
    [
      "0.896207890",
      "0.428466392",
      "0.162493836"
    ],
    [
      "1.239776010",
      "1.529085088",
      "0.162493811"
    ],
    [
      "1.517148448",
      "0.671652217",
      "0.159332757"
    ],
    [
      "1.612103897",
      "0.975841581",
      "0.159332750"
    ],
    [
      "0.495987700",
      "1.157332109",
      "0.157879770"
    ],
    [
      "1.098747887",
      "1.242431152",
      "0.156973574"
    ],
    [
      "0.943296782",
      "0.744444350",
      "0.156973585"
    ],
    [
      "0.602402494",
      "1.451443289",
      "0.154890903"
    ],
    [
      "0.416189845",
      "0.854912194",
      "0.154890916"
    ],
    [
      "0.795202449",
      "1.218466111",
      "0.147516418"
    ],
    [
      "0.707288993",
      "0.936835852",
      "0.147516424"
    ],
    [
      "1.814308141",
      "0.745806431",
      "0.146939557"
    ],
    [
      "1.362554487",
      "1.792707219",
      "0.128317436"
    ],
    [
      "0.847212599",
      "0.141812113",
      "0.128317466"
    ],
    [
      "0.422769808",
      "1.661870807",
      "0.121781542"
    ],
    [
      "0.148747825",
      "0.784043136",
      "0.121781557"
    ],
    [
      "1.729408127",
      "0.498309056",
      "0.114714724"
    ],
    [
      "1.885282167",
      "0.997650921",
      "0.114714716"
    ],
    [
      "0.955827174",
      "1.013789019",
      "0.112662393"
    ]
  ]
}

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

Instance parameters

{
  "n": 34
}

Starter answer

{
  "circles": [
    [
      "1.255654784",
      "0.920195108",
      "0.201433794"
    ],
    [
      "1.582741185",
      "1.572355598",
      "0.183190218"
    ],
    [
      "1.153652641",
      "0.197772459",
      "0.183190253"
    ],
    [
      "0.579922085",
      "0.296174502",
      "0.180344104"
    ],
    [
      "1.054915799",
      "1.817814223",
      "0.180344066"
    ],
    [
      "1.777492444",
      "1.269022454",
      "0.177280374"
    ],
    [
      "0.701655130",
      "1.767589400",
      "0.176469096"
    ],
    [
      "1.486380982",
      "0.336448131",
      "0.177280397"
    ],
    [
      "0.317966369",
      "0.538444776",
      "0.176469126"
    ],
    [
      "1.428103928",
      "1.252014525",
      "0.172521861"
    ],
    [
      "0.175823729",
      "1.076826171",
      "0.172250770"
    ],
    [
      "1.208713194",
      "0.549197354",
      "0.172521878"
    ],
    [
      "0.278477817",
      "1.405677877",
      "0.172250762"
    ],
    [
      "0.915536531",
      "1.504377065",
      "0.162685716"
    ],
    [
      "0.643603605",
      "0.633241464",
      "0.162685736"
    ],
    [
      "0.896207890",
      "0.428466392",
      "0.162493836"
    ],
    [
      "1.239776010",
      "1.529085088",
      "0.162493811"
    ],
    [
      "1.517148448",
      "0.671652217",
      "0.159332757"
    ],
    [
      "1.612103897",
      "0.975841581",
      "0.159332750"
    ],
    [
      "0.495987700",
      "1.157332109",
      "0.157879770"
    ],
    [
      "1.098747887",
      "1.242431152",
      "0.156973574"
    ],
    [
      "0.943296782",
      "0.744444350",
      "0.156973585"
    ],
    [
      "0.602402494",
      "1.451443289",
      "0.154890903"
    ],
    [
      "0.416189845",
      "0.854912194",
      "0.154890916"
    ],
    [
      "0.795202449",
      "1.218466111",
      "0.147516418"
    ],
    [
      "0.707288993",
      "0.936835852",
      "0.147516424"
    ],
    [
      "1.814308141",
      "0.745806431",
      "0.146939557"
    ],
    [
      "1.362554487",
      "1.792707219",
      "0.128317436"
    ],
    [
      "0.847212599",
      "0.141812113",
      "0.128317466"
    ],
    [
      "0.422769808",
      "1.661870807",
      "0.121781542"
    ],
    [
      "0.148747825",
      "0.784043136",
      "0.121781557"
    ],
    [
      "1.729408127",
      "0.498309056",
      "0.114714724"
    ],
    [
      "1.885282167",
      "0.997650921",
      "0.114714716"
    ],
    [
      "0.955827174",
      "1.013789019",
      "0.112662393"
    ]
  ]
}

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