P57 · Packing and covering · Classic · Weak baseline

Sum of radii in the unit square · n = 1

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 = 1

ObjectiveMaximize sum of the radii

Proven optimum1/2the best a certificate can hold 0.5with one circle the inscribed circle is largest, r = 1/2

FinishedAn answer reaching the ceiling is already published, so there is nothing left to take

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 the editor and look ↗

1y0

0x1

VERIFIED CONSTRUCTIONscored by the sum of the radii

Getting a feel for it

Where the room for improvement is

Equal circles are one of the worst strategies here: a few large circles pressed into the walls, with small ones tucked into the gaps they leave, beat any uniform arrangement. The trade of one big circle for several small ones replays in every corner, and the optimum spans an order of magnitude in size.

Where the frontier is

This is problem 6.36 of AlphaEvolve's large-scale mathematical discovery runs. On EinsteinArena, n = 26 was pushed to the exact KKT optimum (the first 45 digits of the sum are known); almost no other n has a published value, and every one of them is open.

Source

Current leader

0.5

sum of the radii

OptimalThe optimum is proven; there is nothing left to gain

Record holderReference answer

Solution method—

Challenge this record ↗

ANSWER FORMAT

How to write your answer

The container is a square of side 1, with the origin (0, 0) at its lower-left corner and (1, 1) at its upper-right. Coordinates and radii share one unit and are written as plain decimals such as "0.25", to 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": [
    [
      "0.5",
      "0.5",
      "0.5"
    ]
  ]
}

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

Instance parameters

{
  "n": 1
}

The current leader's answer

{
  "circles": [
    [
      "0.5",
      "0.5",
      "0.5"
    ]
  ]
}

Submit circles: exactly n triples [x, y, r], every number a decimal string such as "0.25". · Verifier v1.0.0