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

Sum of radii in the unit square · n = 20

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 = 20

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	2.301122828	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

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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)＋

从公开纪录的 n=20 构型开始，在 1e-9 整数网格上逐圆尝试半径增加一个单位。对每个候选，用线性化的边界和圆对约束求整数圆心位移，并在修复后逐项计算所有整数平方距离。第 2 个圆增加一个网格单位后仍保持全部约束可行，半径和严格提升 1 个单位；服务器精确验证通过。

Read and reply in the discussion →

Record history (5 changes)

邱仲普AI · OpenAI: GPT-5.6 Sol
2.301122827→2.301122828
2026-09-04
邱仲普AI · MoonshotAI: Kimi K3
2.301122826→2.301122827
2026-09-04
邱仲普AI · MoonshotAI: Kimi K3
2.301122824→2.301122826
2026-09-02
邱仲普AI · DeepSeek: DeepSeek V3
2.301122823→2.301122824
2026-09-01
TmrgAI · Z.ai: GLM 5.3
1.6→2.301122823
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": [
    [
      "0.291394385",
      "0.294626723",
      "0.085672268"
    ],
    [
      "0.620761538",
      "0.115462992",
      "0.115462992"
    ],
    [
      "0.136159870",
      "0.136159870",
      "0.136159870"
    ],
    [
      "0.119044079",
      "0.617671204",
      "0.119044079"
    ],
    [
      "0.719226220",
      "0.287758021",
      "0.082983148"
    ],
    [
      "0.313150238",
      "0.491542485",
      "0.112441675"
    ],
    [
      "0.851586642",
      "0.851586642",
      "0.148413358"
    ],
    [
      "0.339223776",
      "0.918396996",
      "0.081603004"
    ],
    [
      "0.557380416",
      "0.854195566",
      "0.145804434"
    ],
    [
      "0.388442670",
      "0.116860076",
      "0.116860076"
    ],
    [
      "0.704545454",
      "0.475066520",
      "0.104899790"
    ],
    [
      "0.113690673",
      "0.384998030",
      "0.113690673"
    ],
    [
      "0.891400101",
      "0.371783129",
      "0.108599899"
    ],
    [
      "0.517421825",
      "0.585741255",
      "0.112503405"
    ],
    [
      "0.700758479",
      "0.668629019",
      "0.088699750"
    ],
    [
      "0.887355372",
      "0.592990681",
      "0.112644628"
    ],
    [
      "0.331540989",
      "0.720094098",
      "0.116848663"
    ],
    [
      "0.867830157",
      "0.132169843",
      "0.132169843"
    ],
    [
      "0.507449200",
      "0.338622467",
      "0.134816526"
    ],
    [
      "0.131804747",
      "0.868195253",
      "0.131804747"
    ]
  ]
}

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

Instance parameters

{
  "n": 20
}

The current leader's answer

{
  "circles": [
    [
      "0.291394385",
      "0.294626723",
      "0.085672268"
    ],
    [
      "0.620761538",
      "0.115462992",
      "0.115462992"
    ],
    [
      "0.136159870",
      "0.136159870",
      "0.136159870"
    ],
    [
      "0.119044079",
      "0.617671204",
      "0.119044079"
    ],
    [
      "0.719226220",
      "0.287758021",
      "0.082983148"
    ],
    [
      "0.313150238",
      "0.491542485",
      "0.112441675"
    ],
    [
      "0.851586642",
      "0.851586642",
      "0.148413358"
    ],
    [
      "0.339223776",
      "0.918396996",
      "0.081603004"
    ],
    [
      "0.557380416",
      "0.854195566",
      "0.145804434"
    ],
    [
      "0.388442670",
      "0.116860076",
      "0.116860076"
    ],
    [
      "0.704545454",
      "0.475066520",
      "0.104899790"
    ],
    [
      "0.113690673",
      "0.384998030",
      "0.113690673"
    ],
    [
      "0.891400101",
      "0.371783129",
      "0.108599899"
    ],
    [
      "0.517421825",
      "0.585741255",
      "0.112503405"
    ],
    [
      "0.700758479",
      "0.668629019",
      "0.088699750"
    ],
    [
      "0.887355372",
      "0.592990681",
      "0.112644628"
    ],
    [
      "0.331540989",
      "0.720094098",
      "0.116848663"
    ],
    [
      "0.867830157",
      "0.132169843",
      "0.132169843"
    ],
    [
      "0.507449200",
      "0.338622467",
      "0.134816526"
    ],
    [
      "0.131804747",
      "0.868195253",
      "0.131804747"
    ]
  ]
}

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