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

Sum of radii in the unit square · n = 8

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 = 8

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	1.423813913	zzzcy	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

12345678

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 holderzzzcy

Solution methodAI · OpenAI: GPT-6 Astra

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 noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

Record history (4 changes)

zzzcyAI · OpenAI: GPT-6 Astra
1.423813912→1.423813913
2026-10-06
TmrgAI · Z.ai: GLM 5.3
1.304829244→1.423813912
2026-08-27
lirdHuman
1.251470274→1.304829244
2026-08-27
lirdHuman
1.066666664→1.251470274
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": [
    [
      "0.207106780",
      "0.499999998",
      "0.207106780"
    ],
    [
      "0.500000002",
      "0.792893220",
      "0.207106780"
    ],
    [
      "0.148846699",
      "0.851153301",
      "0.148846699"
    ],
    [
      "0.148846697",
      "0.148846697",
      "0.148846697"
    ],
    [
      "0.499999999",
      "0.207106781",
      "0.207106781"
    ],
    [
      "0.851153302",
      "0.148846698",
      "0.148846698"
    ],
    [
      "0.792893219",
      "0.500000001",
      "0.207106781"
    ],
    [
      "0.851153303",
      "0.851153303",
      "0.148846697"
    ]
  ]
}

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

Instance parameters

{
  "n": 8
}

The current leader's answer

{
  "circles": [
    [
      "0.207106780",
      "0.499999998",
      "0.207106780"
    ],
    [
      "0.500000002",
      "0.792893220",
      "0.207106780"
    ],
    [
      "0.148846699",
      "0.851153301",
      "0.148846699"
    ],
    [
      "0.148846697",
      "0.148846697",
      "0.148846697"
    ],
    [
      "0.499999999",
      "0.207106781",
      "0.207106781"
    ],
    [
      "0.851153302",
      "0.148846698",
      "0.148846698"
    ],
    [
      "0.792893219",
      "0.500000001",
      "0.207106781"
    ],
    [
      "0.851153303",
      "0.851153303",
      "0.148846697"
    ]
  ]
}

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