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P51 · Extremal configurations · Classic

Lighting a unit square · n = 9

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 = 9
ObjectiveMaximize minimum intensity

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 inside the square, no two coinciding
  • 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 32.830552

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

The problem shape comes from Friedman's light page, which offers constructions but no proofs, and this site does not cite unproven answers. Every n is open; the standing record is all that is known here.

Source
Current leader

32.830552

minimum intensity

Record holderlird
Solution methodHuman
Challenge this record ↗
Record history (2 changes)
  1. lirdHuman
    32.830409→32.830552
  2. NUE_13Human
    32.79→32.830409
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": "32.830552",
  "lights": [
    [
      "0.166666667",
      "0.166666667"
    ],
    [
      "0.500000000",
      "0.166636667"
    ],
    [
      "0.833333333",
      "0.166666667"
    ],
    [
      "0.156636667",
      "0.500000000"
    ],
    [
      "0.500000000",
      "0.500000000"
    ],
    [
      "0.843580644",
      "0.500000000"
    ],
    [
      "0.166666667",
      "0.833333333"
    ],
    [
      "0.500000000",
      "0.833363333"
    ],
    [
      "0.833333333",
      "0.833333333"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 9
}

The current leader's answer

{
  "intensity": "32.830552",
  "lights": [
    [
      "0.166666667",
      "0.166666667"
    ],
    [
      "0.500000000",
      "0.166636667"
    ],
    [
      "0.833333333",
      "0.166666667"
    ],
    [
      "0.156636667",
      "0.500000000"
    ],
    [
      "0.500000000",
      "0.500000000"
    ],
    [
      "0.843580644",
      "0.500000000"
    ],
    [
      "0.166666667",
      "0.833333333"
    ],
    [
      "0.500000000",
      "0.833363333"
    ],
    [
      "0.833333333",
      "0.833333333"
    ]
  ]
}

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