P07 · Extremal configurations · Classic

Spreading points in a circle

2y0
0x2
VERIFIED CONSTRUCTIONthe closest pair
n = 8Current record · open

Place n points inside a circle of radius 1, maximizing the smallest distance between any two of them.

Formal definition

  • ContainerA circle of radius 1 centred at (1, 1), so both coordinates run from 0 to 2
  • SubmissionExactly n points, no two coinciding
  • ConstraintsEvery point lies inside the circle or on it
  • ObjectiveMake the smallest pairwise distance as large as possible; compared internally by its square, exactly

Getting a feel for it

Where the room for improvement is

Spreading points IS packing equal circles: discs of half the minimum distance around each point must not overlap. Optima are jammed contact structures, and the container shape decides everything.

Where the frontier is

Spreading points in a disc is dual to packing equal circles in it: a point set with spacing d is a packing of radius d/2. Proofs in the cci table transfer; n = 4 is proven here, and the rest open or close as cci does.

Source
ONE 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.

n4
CURRENT RECORD1.414213562
Optimal
Record holderReference answer
Solution method
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n5
CURRENT RECORD1.071128025
Record holderJev Li
Solution methodHuman
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n6
CURRENT RECORD0.933405848
Record holderNUE_13
Solution methodHuman
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n7
CURRENT RECORD0.917218497
Record holderAnonymous
Solution methodHuman
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n8
CURRENT RECORD0.382683432
Record holderFounding benchmark
Solution methodHuman
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n9
CURRENT RECORD0.342020142
Record holderFounding benchmark
Solution methodHuman
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n10
CURRENT RECORD0.309016994
Record holderFounding benchmark
Solution methodHuman
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n11
CURRENT RECORD0.281732556
Record holderFounding benchmark
Solution methodHuman
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n12
CURRENT RECORD0.258819045
Record holderFounding benchmark
Solution methodHuman
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n13
CURRENT RECORD0.239315663
Record holderFounding benchmark
Solution methodHuman
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n14
CURRENT RECORD0.222520933
Record holderFounding benchmark
Solution methodHuman
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n15
CURRENT RECORD0.20791169
Record holderFounding benchmark
Solution methodHuman
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n16
CURRENT RECORD0.195090321
Record holderFounding benchmark
Solution methodHuman
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n17
CURRENT RECORD0.183749517
Record holderFounding benchmark
Solution methodHuman
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n18
CURRENT RECORD0.173648177
Record holderFounding benchmark
Solution methodHuman
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n19
CURRENT RECORD0.164594589
Record holderFounding benchmark
Solution methodHuman
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n20
CURRENT RECORD0.421738225
Record holderNUE_13
Solution methodHuman
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