P51 · Extremal configurations · Classic · Hard

Lighting a unit square · n = 26

Place n unit-brightness lights in the unit square; the intensity at a point is the sum of 1/distance² to each light. Make the darkest point as bright as possible.

Instancen = 26

ObjectiveMaximize minimum intensity

Best known, unproven234.742765public pictured configuration by Satoshi Kishimoto and GPT-5.6 Sol Pro (July 2026), with continuous value 234.759+. The site reconstructed the pictured topology and symmetries; its nine-decimal certificate verifies at 234.742765; optimality is open

Formal definition

ContainerThe unit square: the origin (0, 0) at its lower-left corner, (1, 1) at its upper right
SubmissionThe n light positions, plus the minimum intensity you claim — the verifier accepts only a bound it can prove
ConstraintsAll lights lie inside the square; several lights may share a position, in which case their intensities add
ObjectiveMake the intensity at the darkest point of the square as large as possible

Open the full editor ↗

1y0

1234567891011121314151617181920212223242526

0x1

VERIFIED CONSTRUCTIONthe darkest point is at least 234.745113

Getting a feel for it

Where the room for improvement is

The darkest point hides in saddles between lights and in the corners: brighten it and somewhere else goes dim. Uniform grids are far from optimal; edges and corners need overprovisioning.

Where the frontier is

Friedman's public table gives a best-known construction and value for each n, without optimality proofs. This site reconstructs those figures and shows the intensity its nine-decimal certificate can rigorously prove as the target; matching it does not settle the problem.

Source

Current leader

234.745113

minimum intensity

Ahead of the best known

Record holderlird

Solution methodHuman

Challenge this record ↗

Record history (1 changes)

lirdHuman
234.742765→234.745113
2026-08-29

ANSWER FORMAT

How to write your answer

The container is a square of side 1, origin at the lower-left corner (0, 0) and (1, 1) at the upper right. Coordinates are decimals such as "0.25", to at most nine places. The intensity is not a coordinate: it is a number larger than 1, to at most six decimal places.

Submit lights and intensity. lights are the n positions, written as decimal strings. intensity is the value you claim no point in the square falls below, to at most six decimals — the verifier proves that bound and refuses what it cannot prove, so claim a little under the minimum you actually reach. The intensity at a point is the sum over lights of one over the squared distance to it.

The current leader's answer

{
  "intensity": "234.745113",
  "lights": [
    [
      "0.269984039",
      "0.978777620"
    ],
    [
      "0.730015961",
      "0.978777620"
    ],
    [
      "0.500000000",
      "0.972694746"
    ],
    [
      "0.060222783",
      "0.949221799"
    ],
    [
      "0.939777217",
      "0.949221799"
    ],
    [
      "0.500000000",
      "0.763578118"
    ],
    [
      "0.260720678",
      "0.752850924"
    ],
    [
      "0.739279322",
      "0.752850924"
    ],
    [
      "0.016378459",
      "0.745836827"
    ],
    [
      "0.983621541",
      "0.745836827"
    ],
    [
      "0.015783970",
      "0.500000000"
    ],
    [
      "0.132999525",
      "0.500000000"
    ],
    [
      "0.382418611",
      "0.500000000"
    ],
    [
      "0.617581389",
      "0.500000000"
    ],
    [
      "0.867000475",
      "0.500000000"
    ],
    [
      "0.984216030",
      "0.500000000"
    ],
    [
      "0.016378459",
      "0.254163173"
    ],
    [
      "0.983621541",
      "0.254163173"
    ],
    [
      "0.260720678",
      "0.247149076"
    ],
    [
      "0.739279322",
      "0.247149076"
    ],
    [
      "0.500000000",
      "0.236421882"
    ],
    [
      "0.060222783",
      "0.050778201"
    ],
    [
      "0.939777217",
      "0.050778201"
    ],
    [
      "0.500000000",
      "0.027305254"
    ],
    [
      "0.269984039",
      "0.021222380"
    ],
    [
      "0.730015961",
      "0.021222380"
    ]
  ]
}

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

Instance parameters

{
  "n": 26
}

The current leader's answer

{
  "intensity": "234.745113",
  "lights": [
    [
      "0.269984039",
      "0.978777620"
    ],
    [
      "0.730015961",
      "0.978777620"
    ],
    [
      "0.500000000",
      "0.972694746"
    ],
    [
      "0.060222783",
      "0.949221799"
    ],
    [
      "0.939777217",
      "0.949221799"
    ],
    [
      "0.500000000",
      "0.763578118"
    ],
    [
      "0.260720678",
      "0.752850924"
    ],
    [
      "0.739279322",
      "0.752850924"
    ],
    [
      "0.016378459",
      "0.745836827"
    ],
    [
      "0.983621541",
      "0.745836827"
    ],
    [
      "0.015783970",
      "0.500000000"
    ],
    [
      "0.132999525",
      "0.500000000"
    ],
    [
      "0.382418611",
      "0.500000000"
    ],
    [
      "0.617581389",
      "0.500000000"
    ],
    [
      "0.867000475",
      "0.500000000"
    ],
    [
      "0.984216030",
      "0.500000000"
    ],
    [
      "0.016378459",
      "0.254163173"
    ],
    [
      "0.983621541",
      "0.254163173"
    ],
    [
      "0.260720678",
      "0.247149076"
    ],
    [
      "0.739279322",
      "0.247149076"
    ],
    [
      "0.500000000",
      "0.236421882"
    ],
    [
      "0.060222783",
      "0.050778201"
    ],
    [
      "0.939777217",
      "0.050778201"
    ],
    [
      "0.500000000",
      "0.027305254"
    ],
    [
      "0.269984039",
      "0.021222380"
    ],
    [
      "0.730015961",
      "0.021222380"
    ]
  ]
}

DISCUSSION

Discussion

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