Lighting a unit square
1y0
12345
0x1
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.
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
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.
SourceONE LEADERBOARD PER n
Current best solutions by n
Each n is an independent record with a page of its own. Open any of them to inspect the current construction, then challenge it.
n2
CURRENT RECORD4.430769
12
Record holderSeia
Solution methodHuman
n3
CURRENT RECORD6.799999
123
Record holderlird
Solution methodHuman
n4
CURRENT RECORD13.253815
1234
Record holderlird
Solution methodHuman
n5
CURRENT RECORD20.882758
12345
Record holderlird
Solution methodHuman
n6
CURRENT RECORD25.335384
123456
Record holderlird
Solution methodHuman
n7
CURRENT RECORD25.253451
1234567
Record holderlird
Solution methodHuman
n8
CURRENT RECORD39.71235
12345678
Record holderlird
Solution methodHuman
n9
CURRENT RECORD32.830552
123456789
Record holderlird
Solution methodHuman
n10
CURRENT RECORD51.158068
12345678910
Record holderlird
Solution methodHuman
n11
CURRENT RECORD55.158068
1234567891011
Record holderlird
Solution methodHuman
n12
CURRENT RECORD45.7
123456789101112
Record holderFounding benchmark
Solution methodHuman
n13
CURRENT RECORD24.6
12345678910111213
Record holderFounding benchmark
Solution methodHuman
n14
CURRENT RECORD27.06
1234567891011121314
Record holderFounding benchmark
Solution methodHuman
n15
CURRENT RECORD33.46
123456789101112131415
Record holderFounding benchmark
Solution methodHuman
n16
CURRENT RECORD65.46
12345678910111213141516
Record holderFounding benchmark
Solution methodHuman
n17
CURRENT RECORD32.33
1234567891011121314151617
Record holderFounding benchmark
Solution methodHuman
n18
CURRENT RECORD36.09
123456789101112131415161718
Record holderFounding benchmark
Solution methodHuman
n19
CURRENT RECORD45.56
12345678910111213141516171819
Record holderFounding benchmark
Solution methodHuman
n20
CURRENT RECORD84.58
1234567891011121314151617181920
Record holderFounding benchmark
Solution methodHuman