MinMax Arena
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P51 · Extremal configurations · Classic · Hard

Lighting a unit square · n = 33

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 = 33
ObjectiveMaximize minimum intensity
Best known, unproven330.591939public pictured configuration by Shaoheng Lai and Codex (August 2026), with continuous value 330.595+. The site reconstructed the pictured topology and symmetries; its nine-decimal certificate verifies at 330.591939; 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 330.595245

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 = 33, 330.595245
Current leader

330.595245

minimum intensity

Ahead of the best known
Record holderlird
Solution methodHuman
Challenge this record ↗
Record history (1 changes)
  1. lirdHuman
    330.591939→330.595245
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": "330.595245",
  "lights": [
    [
      "0.060787397",
      "0.972716714"
    ],
    [
      "0.938839075",
      "0.972716714"
    ],
    [
      "0.243073951",
      "0.970567199"
    ],
    [
      "0.756552521",
      "0.970567199"
    ],
    [
      "0.418067943",
      "0.944320856"
    ],
    [
      "0.581558529",
      "0.944320856"
    ],
    [
      "0.943248751",
      "0.831359747"
    ],
    [
      "0.056377722",
      "0.830612692"
    ],
    [
      "0.280550852",
      "0.773454621"
    ],
    [
      "0.718328565",
      "0.771960510"
    ],
    [
      "0.499439709",
      "0.716122203"
    ],
    [
      "0.946941430",
      "0.670769533"
    ],
    [
      "0.052685042",
      "0.670022478"
    ],
    [
      "0.275526567",
      "0.587654605"
    ],
    [
      "0.724099906",
      "0.587654605"
    ],
    [
      "0.500186764",
      "0.500560291"
    ],
    [
      "0.953674964",
      "0.500560291"
    ],
    [
      "0.045951508",
      "0.499813236"
    ],
    [
      "0.724099906",
      "0.413465978"
    ],
    [
      "0.275526567",
      "0.412718923"
    ],
    [
      "0.052685042",
      "0.330351050"
    ],
    [
      "0.946941430",
      "0.330351050"
    ],
    [
      "0.500186764",
      "0.284251324"
    ],
    [
      "0.281297907",
      "0.228413018"
    ],
    [
      "0.718328565",
      "0.228413018"
    ],
    [
      "0.056377722",
      "0.169760836"
    ],
    [
      "0.943248751",
      "0.169760836"
    ],
    [
      "0.418067943",
      "0.056052672"
    ],
    [
      "0.580811474",
      "0.056052672"
    ],
    [
      "0.243073951",
      "0.030553384"
    ],
    [
      "0.060787397",
      "0.028299281"
    ],
    [
      "0.755805466",
      "0.029806329"
    ],
    [
      "0.938839075",
      "0.027656814"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 33
}

The current leader's answer

{
  "intensity": "330.595245",
  "lights": [
    [
      "0.060787397",
      "0.972716714"
    ],
    [
      "0.938839075",
      "0.972716714"
    ],
    [
      "0.243073951",
      "0.970567199"
    ],
    [
      "0.756552521",
      "0.970567199"
    ],
    [
      "0.418067943",
      "0.944320856"
    ],
    [
      "0.581558529",
      "0.944320856"
    ],
    [
      "0.943248751",
      "0.831359747"
    ],
    [
      "0.056377722",
      "0.830612692"
    ],
    [
      "0.280550852",
      "0.773454621"
    ],
    [
      "0.718328565",
      "0.771960510"
    ],
    [
      "0.499439709",
      "0.716122203"
    ],
    [
      "0.946941430",
      "0.670769533"
    ],
    [
      "0.052685042",
      "0.670022478"
    ],
    [
      "0.275526567",
      "0.587654605"
    ],
    [
      "0.724099906",
      "0.587654605"
    ],
    [
      "0.500186764",
      "0.500560291"
    ],
    [
      "0.953674964",
      "0.500560291"
    ],
    [
      "0.045951508",
      "0.499813236"
    ],
    [
      "0.724099906",
      "0.413465978"
    ],
    [
      "0.275526567",
      "0.412718923"
    ],
    [
      "0.052685042",
      "0.330351050"
    ],
    [
      "0.946941430",
      "0.330351050"
    ],
    [
      "0.500186764",
      "0.284251324"
    ],
    [
      "0.281297907",
      "0.228413018"
    ],
    [
      "0.718328565",
      "0.228413018"
    ],
    [
      "0.056377722",
      "0.169760836"
    ],
    [
      "0.943248751",
      "0.169760836"
    ],
    [
      "0.418067943",
      "0.056052672"
    ],
    [
      "0.580811474",
      "0.056052672"
    ],
    [
      "0.243073951",
      "0.030553384"
    ],
    [
      "0.060787397",
      "0.028299281"
    ],
    [
      "0.755805466",
      "0.029806329"
    ],
    [
      "0.938839075",
      "0.027656814"
    ]
  ]
}

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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