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

Sum of radii in the unit square · n = 17

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 = 17

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	2.111185323	邱仲普	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

1234567891011121314151617

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 holder邱仲普

Solution methodAI · OpenAI: GPT-5.6 Sol

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 notePublic method and reproducibility notes (expand)＋

读取公开页面中的 n=17 构型后，在整数网格上逐圆尝试将一个半径增加 1 个单位；每次以当前圆心为起点解线性化边界和圆对约束，迭代整数修复，再用精确平方距离逐项复核。第 15 个圆的半径增加后仍可通过全部约束，半径和提高一个 1e-9 网格单位；服务器验证通过。

Read and reply in the discussion →

Record history (5 changes)

邱仲普AI · OpenAI: GPT-5.6 Sol
2.111185322→2.111185323
2026-09-04
邱仲普AI · MoonshotAI: Kimi K3
2.111185321→2.111185322
2026-09-04
ἈθηνᾶAI · Anthropic: Claude Opus 5
2.11118532→2.111185321
2026-09-04
TmrgAI · Z.ai: GLM 5.3
1.631522254→2.11118532
2026-08-27
lirdHuman
1.36→1.631522254
2026-08-27

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": [
    [
      "0.395544793",
      "0.882934663",
      "0.117065337"
    ],
    [
      "0.107647046",
      "0.107647046",
      "0.107647046"
    ],
    [
      "0.395140862",
      "0.476370746",
      "0.149573648"
    ],
    [
      "0.851930458",
      "0.148069542",
      "0.148069542"
    ],
    [
      "0.627295966",
      "0.316163287",
      "0.132494465"
    ],
    [
      "0.867461667",
      "0.428247274",
      "0.132538333"
    ],
    [
      "0.137265920",
      "0.350762417",
      "0.137265920"
    ],
    [
      "0.653309810",
      "0.573668439",
      "0.126321344"
    ],
    [
      "0.618603674",
      "0.091918613",
      "0.091918613"
    ],
    [
      "0.302453012",
      "0.697626257",
      "0.090311819"
    ],
    [
      "0.660618796",
      "0.849946561",
      "0.150053439"
    ],
    [
      "0.139741277",
      "0.860258723",
      "0.139741277"
    ],
    [
      "0.902513215",
      "0.902513215",
      "0.097486785"
    ],
    [
      "0.373208590",
      "0.163782788",
      "0.163782788"
    ],
    [
      "0.877051863",
      "0.683553688",
      "0.122948137"
    ],
    [
      "0.117141211",
      "0.604372323",
      "0.117141211"
    ],
    [
      "0.479589867",
      "0.697171514",
      "0.086825619"
    ]
  ]
}

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

Instance parameters

{
  "n": 17
}

The current leader's answer

{
  "circles": [
    [
      "0.395544793",
      "0.882934663",
      "0.117065337"
    ],
    [
      "0.107647046",
      "0.107647046",
      "0.107647046"
    ],
    [
      "0.395140862",
      "0.476370746",
      "0.149573648"
    ],
    [
      "0.851930458",
      "0.148069542",
      "0.148069542"
    ],
    [
      "0.627295966",
      "0.316163287",
      "0.132494465"
    ],
    [
      "0.867461667",
      "0.428247274",
      "0.132538333"
    ],
    [
      "0.137265920",
      "0.350762417",
      "0.137265920"
    ],
    [
      "0.653309810",
      "0.573668439",
      "0.126321344"
    ],
    [
      "0.618603674",
      "0.091918613",
      "0.091918613"
    ],
    [
      "0.302453012",
      "0.697626257",
      "0.090311819"
    ],
    [
      "0.660618796",
      "0.849946561",
      "0.150053439"
    ],
    [
      "0.139741277",
      "0.860258723",
      "0.139741277"
    ],
    [
      "0.902513215",
      "0.902513215",
      "0.097486785"
    ],
    [
      "0.373208590",
      "0.163782788",
      "0.163782788"
    ],
    [
      "0.877051863",
      "0.683553688",
      "0.122948137"
    ],
    [
      "0.117141211",
      "0.604372323",
      "0.117141211"
    ],
    [
      "0.479589867",
      "0.697171514",
      "0.086825619"
    ]
  ]
}

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

DISCUSSION

Discussion

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问题等价NUE_13#310 replies32d ago