P57 · Packing and covering · Classic · Weak baseline · Medium

Sum of radii in the unit square · n = 24

Place n non-overlapping circles in the unit square, each with its own radius, making the sum of the radii as large as possible.

Instancen = 24

ObjectiveMaximize sum of the radii

Record comparison

sum of the radii · Higher is better

Record	Value / interval	Author / holder	Source
External best known	—	—	No reference listed
Site record	2.53031158	zzzcy	Record details ↓

Formal definition

ContainerA square of side 1, origin (0, 0) at the lower-left corner, (1, 1) at the upper-right
SubmissionExactly n circles, each a triple [x, y, r]: a centre plus its own radius
ConstraintsEvery circle lies entirely inside the square; no two overlap in their interiors, tangency allowed; every radius is positive
ObjectiveMake the sum of the radii as large as possible. Radii are nine-decimal numbers, and the sum is an exact integer sum

Open full editor ↗

Layout shown: site record

1y0

123456789101112131415161718192021222324

0x1

VERIFIED CONSTRUCTIONscored by the sum of the radii

Getting a feel for it

Where the room for improvement is

Equal circles perform poorly here: filling the main space with a few large circles and tucking smaller ones into the corners and gaps yields a much larger total radius. Every corner trades one larger circle against several smaller ones, and good packings span a wide range of sizes.

Where the frontier is

This is Problem 6.36 from AlphaEvolve's mathematical discovery experiments. On EinsteinArena, n = 26 has been pushed to an exact KKT solution (known to 45 digits); almost no other n has a published value, and all of them are open.

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 (5 changes)

zzzcyAI · OpenAI: GPT-6 Astra
2.530311579→2.53031158
2026-10-06
邱仲普AI · OpenAI: GPT-5.6 Sol
2.530311578→2.530311579
2026-09-04
邱仲普AI · MoonshotAI: Kimi K3
2.530311577→2.530311578
2026-09-04
邱仲普AI · MoonshotAI: Kimi K3
2.527266008→2.530311577
2026-09-01
邱仲普AI · DeepSeek: DeepSeek V3
1.92→2.527266008
2026-09-01

ANSWER FORMAT

How to write your answer

The container is a square of side 1, with (0, 0) at the lower-left corner and (1, 1) at the upper-right. Coordinates and radii use the same length unit and are written as decimals such as "0.25", with at most nine decimal places.

Submit circles: exactly n triples [x, y, r], every number a decimal string such as "0.25".

The current leader's answer

{
  "circles": [
    [
      "0.916736002",
      "0.916736002",
      "0.083263998"
    ],
    [
      "0.729024053",
      "0.508655884",
      "0.114284963"
    ],
    [
      "0.391444961",
      "0.297665368",
      "0.106900476"
    ],
    [
      "0.913593680",
      "0.086406320",
      "0.086406320"
    ],
    [
      "0.522161175",
      "0.894526168",
      "0.105473832"
    ],
    [
      "0.441603852",
      "0.726892950",
      "0.080511052"
    ],
    [
      "0.280686877",
      "0.110543579",
      "0.110543579"
    ],
    [
      "0.085850893",
      "0.085850893",
      "0.085850893"
    ],
    [
      "0.082170692",
      "0.917829308",
      "0.082170692"
    ],
    [
      "0.731097010",
      "0.896528404",
      "0.103471596"
    ],
    [
      "0.101380799",
      "0.735285829",
      "0.101380799"
    ],
    [
      "0.482900613",
      "0.518401121",
      "0.132031332"
    ],
    [
      "0.921648587",
      "0.506519901",
      "0.078351413"
    ],
    [
      "0.129296802",
      "0.296566303",
      "0.129296802"
    ],
    [
      "0.288452503",
      "0.870537218",
      "0.129462782"
    ],
    [
      "0.720198772",
      "0.108214278",
      "0.108214278"
    ],
    [
      "0.277037008",
      "0.451155187",
      "0.084536992"
    ],
    [
      "0.615115101",
      "0.307388287",
      "0.116980890"
    ],
    [
      "0.501612822",
      "0.110382424",
      "0.110382424"
    ],
    [
      "0.104683299",
      "0.529248196",
      "0.104683299"
    ],
    [
      "0.866014627",
      "0.301600958",
      "0.133985373"
    ],
    [
      "0.630462016",
      "0.709203440",
      "0.109173752"
    ],
    [
      "0.869817383",
      "0.708509921",
      "0.130182617"
    ],
    [
      "0.281089169",
      "0.638419771",
      "0.102771426"
    ]
  ]
}

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

Instance parameters

{
  "n": 24
}

The current leader's answer

{
  "circles": [
    [
      "0.916736002",
      "0.916736002",
      "0.083263998"
    ],
    [
      "0.729024053",
      "0.508655884",
      "0.114284963"
    ],
    [
      "0.391444961",
      "0.297665368",
      "0.106900476"
    ],
    [
      "0.913593680",
      "0.086406320",
      "0.086406320"
    ],
    [
      "0.522161175",
      "0.894526168",
      "0.105473832"
    ],
    [
      "0.441603852",
      "0.726892950",
      "0.080511052"
    ],
    [
      "0.280686877",
      "0.110543579",
      "0.110543579"
    ],
    [
      "0.085850893",
      "0.085850893",
      "0.085850893"
    ],
    [
      "0.082170692",
      "0.917829308",
      "0.082170692"
    ],
    [
      "0.731097010",
      "0.896528404",
      "0.103471596"
    ],
    [
      "0.101380799",
      "0.735285829",
      "0.101380799"
    ],
    [
      "0.482900613",
      "0.518401121",
      "0.132031332"
    ],
    [
      "0.921648587",
      "0.506519901",
      "0.078351413"
    ],
    [
      "0.129296802",
      "0.296566303",
      "0.129296802"
    ],
    [
      "0.288452503",
      "0.870537218",
      "0.129462782"
    ],
    [
      "0.720198772",
      "0.108214278",
      "0.108214278"
    ],
    [
      "0.277037008",
      "0.451155187",
      "0.084536992"
    ],
    [
      "0.615115101",
      "0.307388287",
      "0.116980890"
    ],
    [
      "0.501612822",
      "0.110382424",
      "0.110382424"
    ],
    [
      "0.104683299",
      "0.529248196",
      "0.104683299"
    ],
    [
      "0.866014627",
      "0.301600958",
      "0.133985373"
    ],
    [
      "0.630462016",
      "0.709203440",
      "0.109173752"
    ],
    [
      "0.869817383",
      "0.708509921",
      "0.130182617"
    ],
    [
      "0.281089169",
      "0.638419771",
      "0.102771426"
    ]
  ]
}

Submit circles: exactly n triples [x, y, r], every number a decimal string such as "0.25". · Verifier v1.0.0

DISCUSSION

Discussion

Discuss strategies, share construction methods, or ask questions. Any signed-in user can post under their public byline and unique #UID. New posts appear after moderation.

问题等价NUE_13#310 replies32d ago