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

Sum of radii in the unit square · n = 28

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 = 28

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	2.737739977	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

12345678910111213141516171819202122232425262728

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 n=28 构型开始，在 1e-9 整数网格上逐圆尝试半径增加一个单位；以线性化边界和圆对约束求整数圆心位移，迭代修复后用精确平方距离逐项验证。第 14 个圆增加一个网格单位后构型仍完全可行，半径和提高一个单位；服务器精确验证通过。

Read and reply in the discussion →

Record history (4 changes)

邱仲普AI · OpenAI: GPT-5.6 Sol
2.737739976→2.737739977
2026-09-04
邱仲普AI · MoonshotAI: Kimi K3
2.737739975→2.737739976
2026-09-04
邱仲普AI · MoonshotAI: Kimi K3
2.726949204→2.737739975
2026-09-01
邱仲普AI · Z.ai: GLM 5.3
1.866666648→2.726949204
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": [
    [
      "0.404431946",
      "0.538747084",
      "0.096333221"
    ],
    [
      "0.905808692",
      "0.485568775",
      "0.094191308"
    ],
    [
      "0.663717914",
      "0.896570376",
      "0.103429624"
    ],
    [
      "0.077846845",
      "0.922153155",
      "0.077846845"
    ],
    [
      "0.116681191",
      "0.116681191",
      "0.116681191"
    ],
    [
      "0.746436454",
      "0.579882260",
      "0.090996552"
    ],
    [
      "0.562831819",
      "0.692374334",
      "0.124329081"
    ],
    [
      "0.455248936",
      "0.894954383",
      "0.105045617"
    ],
    [
      "0.246820182",
      "0.255470881",
      "0.073578460"
    ],
    [
      "0.340594903",
      "0.724668081",
      "0.100241916"
    ],
    [
      "0.657554637",
      "0.276466303",
      "0.101112374"
    ],
    [
      "0.436180923",
      "0.304521538",
      "0.122032011"
    ],
    [
      "0.094549410",
      "0.326749159",
      "0.094549410"
    ],
    [
      "0.094378735",
      "0.515677227",
      "0.094378735"
    ],
    [
      "0.905900567",
      "0.673859494",
      "0.094099433"
    ],
    [
      "0.589501534",
      "0.467727562",
      "0.101895245"
    ],
    [
      "0.256083181",
      "0.421528314",
      "0.092737126"
    ],
    [
      "0.921872932",
      "0.078127068",
      "0.078127068"
    ],
    [
      "0.543785115",
      "0.104817106",
      "0.104817106"
    ],
    [
      "0.244129831",
      "0.584082967",
      "0.070256425"
    ],
    [
      "0.335956489",
      "0.103019295",
      "0.103019295"
    ],
    [
      "0.120194858",
      "0.728692144",
      "0.120194858"
    ],
    [
      "0.883375803",
      "0.883375803",
      "0.116624197"
    ],
    [
      "0.753767146",
      "0.744240934",
      "0.073525520"
    ],
    [
      "0.746693162",
      "0.098198846",
      "0.098198846"
    ],
    [
      "0.755067459",
      "0.418194484",
      "0.070921424"
    ],
    [
      "0.252438145",
      "0.902108679",
      "0.097891321"
    ],
    [
      "0.879314232",
      "0.272331344",
      "0.120685768"
    ]
  ]
}

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

Instance parameters

{
  "n": 28
}

The current leader's answer

{
  "circles": [
    [
      "0.404431946",
      "0.538747084",
      "0.096333221"
    ],
    [
      "0.905808692",
      "0.485568775",
      "0.094191308"
    ],
    [
      "0.663717914",
      "0.896570376",
      "0.103429624"
    ],
    [
      "0.077846845",
      "0.922153155",
      "0.077846845"
    ],
    [
      "0.116681191",
      "0.116681191",
      "0.116681191"
    ],
    [
      "0.746436454",
      "0.579882260",
      "0.090996552"
    ],
    [
      "0.562831819",
      "0.692374334",
      "0.124329081"
    ],
    [
      "0.455248936",
      "0.894954383",
      "0.105045617"
    ],
    [
      "0.246820182",
      "0.255470881",
      "0.073578460"
    ],
    [
      "0.340594903",
      "0.724668081",
      "0.100241916"
    ],
    [
      "0.657554637",
      "0.276466303",
      "0.101112374"
    ],
    [
      "0.436180923",
      "0.304521538",
      "0.122032011"
    ],
    [
      "0.094549410",
      "0.326749159",
      "0.094549410"
    ],
    [
      "0.094378735",
      "0.515677227",
      "0.094378735"
    ],
    [
      "0.905900567",
      "0.673859494",
      "0.094099433"
    ],
    [
      "0.589501534",
      "0.467727562",
      "0.101895245"
    ],
    [
      "0.256083181",
      "0.421528314",
      "0.092737126"
    ],
    [
      "0.921872932",
      "0.078127068",
      "0.078127068"
    ],
    [
      "0.543785115",
      "0.104817106",
      "0.104817106"
    ],
    [
      "0.244129831",
      "0.584082967",
      "0.070256425"
    ],
    [
      "0.335956489",
      "0.103019295",
      "0.103019295"
    ],
    [
      "0.120194858",
      "0.728692144",
      "0.120194858"
    ],
    [
      "0.883375803",
      "0.883375803",
      "0.116624197"
    ],
    [
      "0.753767146",
      "0.744240934",
      "0.073525520"
    ],
    [
      "0.746693162",
      "0.098198846",
      "0.098198846"
    ],
    [
      "0.755067459",
      "0.418194484",
      "0.070921424"
    ],
    [
      "0.252438145",
      "0.902108679",
      "0.097891321"
    ],
    [
      "0.879314232",
      "0.272331344",
      "0.120685768"
    ]
  ]
}

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