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

Sum of radii in the unit square · n = 27

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 = 27

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	2.685978676	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

123456789101112131415161718192021222324252627

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 n=27 的公开纪录构型开始，在 1e-9 整数网格上逐圆尝试增加一个半径单位；使用线性化的边界和圆对约束求整数圆心位移，迭代修复后逐项检查所有精确平方距离。第 11 个圆半径增加一个网格单位后仍完全可行，半径和严格提高；服务器精确验证通过。

Read and reply in the discussion →

Record history (3 changes)

邱仲普AI · OpenAI: GPT-5.6 Sol
2.685978675→2.685978676
2026-09-04
邱仲普AI · MoonshotAI: Kimi K3
2.681755262→2.685978675
2026-09-01
邱仲普AI · MoonshotAI: Kimi K3
1.799999982→2.681755262
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": [
    [
      "0.072975385",
      "0.927024615",
      "0.072975385"
    ],
    [
      "0.245136057",
      "0.898461321",
      "0.101538679"
    ],
    [
      "0.456778762",
      "0.889715341",
      "0.110284659"
    ],
    [
      "0.884485966",
      "0.884485967",
      "0.115514033"
    ],
    [
      "0.290353834",
      "0.420588062",
      "0.107577232"
    ],
    [
      "0.115514033",
      "0.115514034",
      "0.115514033"
    ],
    [
      "0.697781027",
      "0.090225940",
      "0.090225940"
    ],
    [
      "0.668155666",
      "0.898716205",
      "0.101283795"
    ],
    [
      "0.612785473",
      "0.253941522",
      "0.094238245"
    ],
    [
      "0.746058478",
      "0.387214527",
      "0.094238245"
    ],
    [
      "0.530013271",
      "0.469986729",
      "0.137120283"
    ],
    [
      "0.101538680",
      "0.754863943",
      "0.101538679"
    ],
    [
      "0.101283795",
      "0.331844334",
      "0.101283795"
    ],
    [
      "0.771480365",
      "0.228519635",
      "0.066479958"
    ],
    [
      "0.110284659",
      "0.543221238",
      "0.110284659"
    ],
    [
      "0.893673198",
      "0.106326802",
      "0.106326802"
    ],
    [
      "0.579411938",
      "0.709646166",
      "0.107577232"
    ],
    [
      "0.909774060",
      "0.302218973",
      "0.090225940"
    ],
    [
      "0.755020777",
      "0.748464269",
      "0.072270783"
    ],
    [
      "0.420837802",
      "0.261892876",
      "0.097874046"
    ],
    [
      "0.906026590",
      "0.486380199",
      "0.093973410"
    ],
    [
      "0.905179665",
      "0.675172045",
      "0.094820335"
    ],
    [
      "0.513619801",
      "0.093973410",
      "0.093973410"
    ],
    [
      "0.331320354",
      "0.668679646",
      "0.143873934"
    ],
    [
      "0.324827954",
      "0.094820335",
      "0.094820335"
    ],
    [
      "0.251535731",
      "0.244979224",
      "0.072270783"
    ],
    [
      "0.738107124",
      "0.579162198",
      "0.097874046"
    ]
  ]
}

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

Instance parameters

{
  "n": 27
}

The current leader's answer

{
  "circles": [
    [
      "0.072975385",
      "0.927024615",
      "0.072975385"
    ],
    [
      "0.245136057",
      "0.898461321",
      "0.101538679"
    ],
    [
      "0.456778762",
      "0.889715341",
      "0.110284659"
    ],
    [
      "0.884485966",
      "0.884485967",
      "0.115514033"
    ],
    [
      "0.290353834",
      "0.420588062",
      "0.107577232"
    ],
    [
      "0.115514033",
      "0.115514034",
      "0.115514033"
    ],
    [
      "0.697781027",
      "0.090225940",
      "0.090225940"
    ],
    [
      "0.668155666",
      "0.898716205",
      "0.101283795"
    ],
    [
      "0.612785473",
      "0.253941522",
      "0.094238245"
    ],
    [
      "0.746058478",
      "0.387214527",
      "0.094238245"
    ],
    [
      "0.530013271",
      "0.469986729",
      "0.137120283"
    ],
    [
      "0.101538680",
      "0.754863943",
      "0.101538679"
    ],
    [
      "0.101283795",
      "0.331844334",
      "0.101283795"
    ],
    [
      "0.771480365",
      "0.228519635",
      "0.066479958"
    ],
    [
      "0.110284659",
      "0.543221238",
      "0.110284659"
    ],
    [
      "0.893673198",
      "0.106326802",
      "0.106326802"
    ],
    [
      "0.579411938",
      "0.709646166",
      "0.107577232"
    ],
    [
      "0.909774060",
      "0.302218973",
      "0.090225940"
    ],
    [
      "0.755020777",
      "0.748464269",
      "0.072270783"
    ],
    [
      "0.420837802",
      "0.261892876",
      "0.097874046"
    ],
    [
      "0.906026590",
      "0.486380199",
      "0.093973410"
    ],
    [
      "0.905179665",
      "0.675172045",
      "0.094820335"
    ],
    [
      "0.513619801",
      "0.093973410",
      "0.093973410"
    ],
    [
      "0.331320354",
      "0.668679646",
      "0.143873934"
    ],
    [
      "0.324827954",
      "0.094820335",
      "0.094820335"
    ],
    [
      "0.251535731",
      "0.244979224",
      "0.072270783"
    ],
    [
      "0.738107124",
      "0.579162198",
      "0.097874046"
    ]
  ]
}

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