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P51 · Extremal configurations · Classic · Hard

Lighting a unit square · n = 35

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 = 35
ObjectiveMaximize minimum intensity
Best known, unproven347.191734public pictured configuration by Shaoheng Lai and Codex (August 2026), with continuous value 347.195+. The site reconstructed the pictured topology and symmetries; its nine-decimal certificate verifies at 347.191734; 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
0x1
VERIFIED CONSTRUCTIONthe darkest point is at least 347.195206

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
The record-holding arrangement for Lighting a unit square n = 35, 347.195206
Current leader

347.195206

minimum intensity

Ahead of the best known
Record holderlird
Solution methodHuman
Challenge this record ↗
Record history (1 changes)
  1. lirdHuman
    347.191734→347.195206
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": "347.195206",
  "lights": [
    [
      "0.055412914",
      "0.972909248"
    ],
    [
      "0.943605845",
      "0.972909248"
    ],
    [
      "0.231387538",
      "0.960152213"
    ],
    [
      "0.768121841",
      "0.959661593"
    ],
    [
      "0.404512521",
      "0.954023535"
    ],
    [
      "0.594996858",
      "0.954023535"
    ],
    [
      "0.499754690",
      "0.866582841"
    ],
    [
      "0.941938232",
      "0.834828617"
    ],
    [
      "0.057080527",
      "0.834337997"
    ],
    [
      "0.265506014",
      "0.750086490"
    ],
    [
      "0.733022124",
      "0.749595870"
    ],
    [
      "0.959372288",
      "0.675186328"
    ],
    [
      "0.498773449",
      "0.675608207"
    ],
    [
      "0.040137091",
      "0.674205087"
    ],
    [
      "0.751850438",
      "0.565081919"
    ],
    [
      "0.247658941",
      "0.563610057"
    ],
    [
      "0.500735931",
      "0.533517738"
    ],
    [
      "0.962386875",
      "0.501418066"
    ],
    [
      "0.036631884",
      "0.500927445"
    ],
    [
      "0.274601971",
      "0.420122416"
    ],
    [
      "0.725398029",
      "0.419141175"
    ],
    [
      "0.947498413",
      "0.330883979"
    ],
    [
      "0.052501587",
      "0.330393358"
    ],
    [
      "0.499264069",
      "0.320847410"
    ],
    [
      "0.264921303",
      "0.229849397"
    ],
    [
      "0.735078697",
      "0.229849397"
    ],
    [
      "0.049721585",
      "0.169546454"
    ],
    [
      "0.949787794",
      "0.169055834"
    ],
    [
      "0.501226551",
      "0.118923926"
    ],
    [
      "0.407875721",
      "0.054421208"
    ],
    [
      "0.590652417",
      "0.053439967"
    ],
    [
      "0.233566392",
      "0.034114785"
    ],
    [
      "0.765942987",
      "0.034114785"
    ],
    [
      "0.056731568",
      "0.025723974"
    ],
    [
      "0.942777811",
      "0.025302041"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 35
}

The current leader's answer

{
  "intensity": "347.195206",
  "lights": [
    [
      "0.055412914",
      "0.972909248"
    ],
    [
      "0.943605845",
      "0.972909248"
    ],
    [
      "0.231387538",
      "0.960152213"
    ],
    [
      "0.768121841",
      "0.959661593"
    ],
    [
      "0.404512521",
      "0.954023535"
    ],
    [
      "0.594996858",
      "0.954023535"
    ],
    [
      "0.499754690",
      "0.866582841"
    ],
    [
      "0.941938232",
      "0.834828617"
    ],
    [
      "0.057080527",
      "0.834337997"
    ],
    [
      "0.265506014",
      "0.750086490"
    ],
    [
      "0.733022124",
      "0.749595870"
    ],
    [
      "0.959372288",
      "0.675186328"
    ],
    [
      "0.498773449",
      "0.675608207"
    ],
    [
      "0.040137091",
      "0.674205087"
    ],
    [
      "0.751850438",
      "0.565081919"
    ],
    [
      "0.247658941",
      "0.563610057"
    ],
    [
      "0.500735931",
      "0.533517738"
    ],
    [
      "0.962386875",
      "0.501418066"
    ],
    [
      "0.036631884",
      "0.500927445"
    ],
    [
      "0.274601971",
      "0.420122416"
    ],
    [
      "0.725398029",
      "0.419141175"
    ],
    [
      "0.947498413",
      "0.330883979"
    ],
    [
      "0.052501587",
      "0.330393358"
    ],
    [
      "0.499264069",
      "0.320847410"
    ],
    [
      "0.264921303",
      "0.229849397"
    ],
    [
      "0.735078697",
      "0.229849397"
    ],
    [
      "0.049721585",
      "0.169546454"
    ],
    [
      "0.949787794",
      "0.169055834"
    ],
    [
      "0.501226551",
      "0.118923926"
    ],
    [
      "0.407875721",
      "0.054421208"
    ],
    [
      "0.590652417",
      "0.053439967"
    ],
    [
      "0.233566392",
      "0.034114785"
    ],
    [
      "0.765942987",
      "0.034114785"
    ],
    [
      "0.056731568",
      "0.025723974"
    ],
    [
      "0.942777811",
      "0.025302041"
    ]
  ]
}

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. · Verifier v1.0.0

DISCUSSION

Discussion

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