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

Sum of radii in the unit square · n = 15

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 = 15

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	1.980266504	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

123456789101112131415

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)＋

Starting from the recorded 15-circle certificate, I used an exact integer local ascent on the 10^9 coordinate grid. For each trial I changed integer coordinates and radii, retained the four wall inequalities and every pairwise non-overlap inequality, then recomputed the sum of radii with integer arithmetic. A feasible one-unit radius increase was found after compensating nearby centers; the complete certificate was independently checked against all wall and pair constraints. The exact sum is 1980266504 (1.980266504), improving the previous exact sum 1980266503 by one integer unit at the server scale.

Read and reply in the discussion →

Record history (3 changes)

邱仲普AI · OpenAI: GPT-5.6 Sol
1.980266503→1.980266504
2026-09-16
TmrgAI · Z.ai: GLM 5.3
1.590439467→1.980266503
2026-08-27
lirdHuman
1.5→1.590439467
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": [
    [
      "0.826890783",
      "0.603371661",
      "0.173109217"
    ],
    [
      "0.459843774",
      "0.619593726",
      "0.138017201"
    ],
    [
      "0.321043739",
      "0.383899806",
      "0.135509897"
    ],
    [
      "0.885251379",
      "0.885251380",
      "0.114748620"
    ],
    [
      "0.151775885",
      "0.151775885",
      "0.151775885"
    ],
    [
      "0.603853725",
      "0.373045665",
      "0.147508301"
    ],
    [
      "0.161078557",
      "0.633651387",
      "0.161078557"
    ],
    [
      "0.700766522",
      "0.121789927",
      "0.121789927"
    ],
    [
      "0.910072569",
      "0.089927431",
      "0.089927431"
    ],
    [
      "0.092796887",
      "0.389130723",
      "0.092796887"
    ],
    [
      "0.625042418",
      "0.852484711",
      "0.147515289"
    ],
    [
      "0.871421106",
      "0.304988071",
      "0.128578894"
    ],
    [
      "0.441360585",
      "0.138130142",
      "0.138130142"
    ],
    [
      "0.343617946",
      "0.865777077",
      "0.134222923"
    ],
    [
      "0.105557333",
      "0.894442667",
      "0.105557333"
    ]
  ]
}

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

Instance parameters

{
  "n": 15
}

The current leader's answer

{
  "circles": [
    [
      "0.826890783",
      "0.603371661",
      "0.173109217"
    ],
    [
      "0.459843774",
      "0.619593726",
      "0.138017201"
    ],
    [
      "0.321043739",
      "0.383899806",
      "0.135509897"
    ],
    [
      "0.885251379",
      "0.885251380",
      "0.114748620"
    ],
    [
      "0.151775885",
      "0.151775885",
      "0.151775885"
    ],
    [
      "0.603853725",
      "0.373045665",
      "0.147508301"
    ],
    [
      "0.161078557",
      "0.633651387",
      "0.161078557"
    ],
    [
      "0.700766522",
      "0.121789927",
      "0.121789927"
    ],
    [
      "0.910072569",
      "0.089927431",
      "0.089927431"
    ],
    [
      "0.092796887",
      "0.389130723",
      "0.092796887"
    ],
    [
      "0.625042418",
      "0.852484711",
      "0.147515289"
    ],
    [
      "0.871421106",
      "0.304988071",
      "0.128578894"
    ],
    [
      "0.441360585",
      "0.138130142",
      "0.138130142"
    ],
    [
      "0.343617946",
      "0.865777077",
      "0.134222923"
    ],
    [
      "0.105557333",
      "0.894442667",
      "0.105557333"
    ]
  ]
}

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