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P92 · Packing and covering · Classic · Hard

Sum of radii in the unit circle · n = 44

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 = 44
ObjectiveMaximize the sum of the radii

Record comparison

the sum of the radii · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[6.113925, 6.113935]See source attributionErich Friedman ↗
Site recordNo 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 betterS=Σr
  • ObjectiveMaximise the sum of the radii over every legal arrangementmaxSΣr
Open the full editor ↗

Layout shown: starter layout

2y0
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
The current arrangement for Sum of radii in the unit circle n = 44
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": [
    [
      "1.183909821",
      "1.776324589",
      "0.202188812"
    ],
    [
      "0.933764218",
      "0.713391747",
      "0.184626460"
    ],
    [
      "0.690485212",
      "1.194504066",
      "0.164964770"
    ],
    [
      "0.563775072",
      "0.279072524",
      "0.157368043"
    ],
    [
      "1.677949912",
      "1.500652004",
      "0.157225705"
    ],
    [
      "1.054204120",
      "0.157803051",
      "0.156060554"
    ],
    [
      "0.619263727",
      "0.586117931",
      "0.154650982"
    ],
    [
      "1.149287882",
      "0.452603373",
      "0.153694431"
    ],
    [
      "0.329429931",
      "0.482694705",
      "0.153082656"
    ],
    [
      "0.851804076",
      "0.388270870",
      "0.150666022"
    ],
    [
      "1.352568329",
      "0.226752286",
      "0.150166808"
    ],
    [
      "1.265963385",
      "0.730857546",
      "0.148031533"
    ],
    [
      "0.641226733",
      "1.773641287",
      "0.147216851"
    ],
    [
      "1.373460595",
      "1.482073718",
      "0.147829858"
    ],
    [
      "0.412508118",
      "1.055030389",
      "0.146040328"
    ],
    [
      "0.149450782",
      "0.927373508",
      "0.146355705"
    ],
    [
      "0.650949944",
      "0.885578008",
      "0.146480805"
    ],
    [
      "1.821549016",
      "1.233766077",
      "0.145839964"
    ],
    [
      "1.532055360",
      "1.237483703",
      "0.143677559"
    ],
    [
      "0.171847278",
      "1.216802959",
      "0.143938989"
    ],
    [
      "0.400368245",
      "1.614177673",
      "0.141645495"
    ],
    [
      "1.713820840",
      "0.522585830",
      "0.141242477"
    ],
    [
      "0.388742107",
      "0.769847209",
      "0.140131419"
    ],
    [
      "1.552946007",
      "0.753037693",
      "0.139806935"
    ],
    [
      "1.398675013",
      "0.986805265",
      "0.140276654"
    ],
    [
      "1.831221898",
      "0.777664395",
      "0.139556529"
    ],
    [
      "1.249534131",
      "1.223168568",
      "0.139206105"
    ],
    [
      "1.120021872",
      "0.977474873",
      "0.138532650"
    ],
    [
      "1.089498449",
      "1.449505472",
      "0.137993846"
    ],
    [
      "1.436073702",
      "0.501803460",
      "0.137281092"
    ],
    [
      "0.425030595",
      "1.337210513",
      "0.136417517"
    ],
    [
      "0.648339134",
      "1.491503708",
      "0.135010361"
    ],
    [
      "0.881945012",
      "1.626611635",
      "0.134852303"
    ],
    [
      "1.671684159",
      "0.998636401",
      "0.132988728"
    ],
    [
      "0.983056301",
      "1.205924039",
      "0.127829112"
    ],
    [
      "1.503335258",
      "1.718982554",
      "0.122342723"
    ],
    [
      "0.882682160",
      "1.876824595",
      "0.115361742"
    ],
    [
      "0.789141377",
      "0.135293928",
      "0.109956208"
    ],
    [
      "0.876660034",
      "0.998043732",
      "0.105696876"
    ],
    [
      "0.163202714",
      "0.678006660",
      "0.103390046"
    ],
    [
      "1.586098222",
      "0.317489924",
      "0.100371671"
    ],
    [
      "0.221277894",
      "1.454165203",
      "0.098515585"
    ],
    [
      "0.860151936",
      "1.395054793",
      "0.097727809"
    ],
    [
      "1.902300641",
      "1.004013536",
      "0.097690432"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 44
}

Starter answer

{
  "circles": [
    [
      "1.183909821",
      "1.776324589",
      "0.202188812"
    ],
    [
      "0.933764218",
      "0.713391747",
      "0.184626460"
    ],
    [
      "0.690485212",
      "1.194504066",
      "0.164964770"
    ],
    [
      "0.563775072",
      "0.279072524",
      "0.157368043"
    ],
    [
      "1.677949912",
      "1.500652004",
      "0.157225705"
    ],
    [
      "1.054204120",
      "0.157803051",
      "0.156060554"
    ],
    [
      "0.619263727",
      "0.586117931",
      "0.154650982"
    ],
    [
      "1.149287882",
      "0.452603373",
      "0.153694431"
    ],
    [
      "0.329429931",
      "0.482694705",
      "0.153082656"
    ],
    [
      "0.851804076",
      "0.388270870",
      "0.150666022"
    ],
    [
      "1.352568329",
      "0.226752286",
      "0.150166808"
    ],
    [
      "1.265963385",
      "0.730857546",
      "0.148031533"
    ],
    [
      "0.641226733",
      "1.773641287",
      "0.147216851"
    ],
    [
      "1.373460595",
      "1.482073718",
      "0.147829858"
    ],
    [
      "0.412508118",
      "1.055030389",
      "0.146040328"
    ],
    [
      "0.149450782",
      "0.927373508",
      "0.146355705"
    ],
    [
      "0.650949944",
      "0.885578008",
      "0.146480805"
    ],
    [
      "1.821549016",
      "1.233766077",
      "0.145839964"
    ],
    [
      "1.532055360",
      "1.237483703",
      "0.143677559"
    ],
    [
      "0.171847278",
      "1.216802959",
      "0.143938989"
    ],
    [
      "0.400368245",
      "1.614177673",
      "0.141645495"
    ],
    [
      "1.713820840",
      "0.522585830",
      "0.141242477"
    ],
    [
      "0.388742107",
      "0.769847209",
      "0.140131419"
    ],
    [
      "1.552946007",
      "0.753037693",
      "0.139806935"
    ],
    [
      "1.398675013",
      "0.986805265",
      "0.140276654"
    ],
    [
      "1.831221898",
      "0.777664395",
      "0.139556529"
    ],
    [
      "1.249534131",
      "1.223168568",
      "0.139206105"
    ],
    [
      "1.120021872",
      "0.977474873",
      "0.138532650"
    ],
    [
      "1.089498449",
      "1.449505472",
      "0.137993846"
    ],
    [
      "1.436073702",
      "0.501803460",
      "0.137281092"
    ],
    [
      "0.425030595",
      "1.337210513",
      "0.136417517"
    ],
    [
      "0.648339134",
      "1.491503708",
      "0.135010361"
    ],
    [
      "0.881945012",
      "1.626611635",
      "0.134852303"
    ],
    [
      "1.671684159",
      "0.998636401",
      "0.132988728"
    ],
    [
      "0.983056301",
      "1.205924039",
      "0.127829112"
    ],
    [
      "1.503335258",
      "1.718982554",
      "0.122342723"
    ],
    [
      "0.882682160",
      "1.876824595",
      "0.115361742"
    ],
    [
      "0.789141377",
      "0.135293928",
      "0.109956208"
    ],
    [
      "0.876660034",
      "0.998043732",
      "0.105696876"
    ],
    [
      "0.163202714",
      "0.678006660",
      "0.103390046"
    ],
    [
      "1.586098222",
      "0.317489924",
      "0.100371671"
    ],
    [
      "0.221277894",
      "1.454165203",
      "0.098515585"
    ],
    [
      "0.860151936",
      "1.395054793",
      "0.097727809"
    ],
    [
      "1.902300641",
      "1.004013536",
      "0.097690432"
    ]
  ]
}

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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