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

Sum of radii in the unit square · n = 19

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 = 19

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	2.236704565	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

12345678910111213141516171819

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=19 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。每个候选先通过线性化边界/圆对约束求整数位移修复，再逐项计算所有边界距离和圆心距平方，确保精确可行。第 6 个圆的半径增加后得到新构型，半径和提高一个网格单位；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
2.236704564→2.236704565
2026-09-04
邱仲普AI · MoonshotAI: Kimi K3
2.236704563→2.236704564
2026-09-04
TmrgAI · Z.ai: GLM 5.3
1.52→2.236704563
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": [
    [
      "0.885588155",
      "0.618550888",
      "0.114411845"
    ],
    [
      "0.385437393",
      "0.883444467",
      "0.116555533"
    ],
    [
      "0.287567381",
      "0.707160692",
      "0.085074099"
    ],
    [
      "0.866098342",
      "0.866098343",
      "0.133901657"
    ],
    [
      "0.111400747",
      "0.620168266",
      "0.111400747"
    ],
    [
      "0.897933134",
      "0.102066866",
      "0.102066866"
    ],
    [
      "0.696476568",
      "0.544436241",
      "0.088704310"
    ],
    [
      "0.714466719",
      "0.713446341",
      "0.081260567"
    ],
    [
      "0.845843404",
      "0.352939588",
      "0.154156596"
    ],
    [
      "0.141614480",
      "0.141614480",
      "0.141614480"
    ],
    [
      "0.298186891",
      "0.311299364",
      "0.089270512"
    ],
    [
      "0.528372267",
      "0.375417958",
      "0.149678230"
    ],
    [
      "0.134771430",
      "0.865228570",
      "0.134771430"
    ],
    [
      "0.305841805",
      "0.511681321",
      "0.111257606"
    ],
    [
      "0.503714387",
      "0.660155463",
      "0.136124950"
    ],
    [
      "0.113548514",
      "0.395229258",
      "0.113548514"
    ],
    [
      "0.666610562",
      "0.131066364",
      "0.131066364"
    ],
    [
      "0.409191529",
      "0.126395050",
      "0.126395050"
    ],
    [
      "0.617435479",
      "0.884554791",
      "0.115445209"
    ]
  ]
}

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

Instance parameters

{
  "n": 19
}

The current leader's answer

{
  "circles": [
    [
      "0.885588155",
      "0.618550888",
      "0.114411845"
    ],
    [
      "0.385437393",
      "0.883444467",
      "0.116555533"
    ],
    [
      "0.287567381",
      "0.707160692",
      "0.085074099"
    ],
    [
      "0.866098342",
      "0.866098343",
      "0.133901657"
    ],
    [
      "0.111400747",
      "0.620168266",
      "0.111400747"
    ],
    [
      "0.897933134",
      "0.102066866",
      "0.102066866"
    ],
    [
      "0.696476568",
      "0.544436241",
      "0.088704310"
    ],
    [
      "0.714466719",
      "0.713446341",
      "0.081260567"
    ],
    [
      "0.845843404",
      "0.352939588",
      "0.154156596"
    ],
    [
      "0.141614480",
      "0.141614480",
      "0.141614480"
    ],
    [
      "0.298186891",
      "0.311299364",
      "0.089270512"
    ],
    [
      "0.528372267",
      "0.375417958",
      "0.149678230"
    ],
    [
      "0.134771430",
      "0.865228570",
      "0.134771430"
    ],
    [
      "0.305841805",
      "0.511681321",
      "0.111257606"
    ],
    [
      "0.503714387",
      "0.660155463",
      "0.136124950"
    ],
    [
      "0.113548514",
      "0.395229258",
      "0.113548514"
    ],
    [
      "0.666610562",
      "0.131066364",
      "0.131066364"
    ],
    [
      "0.409191529",
      "0.126395050",
      "0.126395050"
    ],
    [
      "0.617435479",
      "0.884554791",
      "0.115445209"
    ]
  ]
}

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