MinMax Arena
中/EN
P34 · Extremal configurations · Classic · Weak baseline

Riesz 2-energy in a disc

2y0
0x2
VERIFIED CONSTRUCTIONthe closest pair
n = 10Current record · open

Place n points inside a circle of radius 1, minimizing the sum of 1/distance² taken over every pair.

Formal definition

  • ContainerThe container is a circle of radius 1 centred at (1, 1), so coordinates run from 0 to 2.
  • SubmissionExactly n points, no two coinciding
  • ConstraintsEvery point lies inside the container or on its boundary
  • ObjectiveMake the sum of 1/distance² over all pairs as small as possible; scored in exact rationals

Getting a feel for it

Where the room for improvement is

1/distance² punishes closeness brutally: points get pushed out to a boundary ring first, then shed inner layers as n grows. Layer counts jump at particular n, and the jumps are where the contest lives.

Where the frontier is

Same theory as the square version: Borodachov, Hardin and Saff (2019); no per-n table of optima in a disc either. The regular-polygon closed forms are trivial cases proved on site; everything else is open.

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.

n5
CURRENT RECORD5.000000008best known 5
Record holderFounding benchmark
Solution methodHuman
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n6
CURRENT RECORD8.750000013best known 8.75
Record holderFounding benchmark
Solution methodHuman
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n7
CURRENT RECORD14.000000026best known 14
Record holderFounding benchmark
Solution methodHuman
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n8
CURRENT RECORD21.000000032best known 21
Record holderFounding benchmark
Solution methodHuman
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n9
CURRENT RECORD29.00000004
Record holderlird
Solution methodHuman
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n10
CURRENT RECORD1385.545299148
Record holderFounding benchmark
Solution methodHuman
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n11
CURRENT RECORD1655.945299148
Record holderFounding benchmark
Solution methodHuman
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n12
CURRENT RECORD1899.979487184
Record holderFounding benchmark
Solution methodHuman
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n13
CURRENT RECORD2078.8923077
Record holderFounding benchmark
Solution methodHuman
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n14
CURRENT RECORD2358.126495736
Record holderFounding benchmark
Solution methodHuman
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n15
CURRENT RECORD2653.360683772
Record holderFounding benchmark
Solution methodHuman
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n16
CURRENT RECORD2919.3846154
Record holderFounding benchmark
Solution methodHuman
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n17
CURRENT RECORD4808.700226262
Record holderFounding benchmark
Solution methodHuman
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n18
CURRENT RECORD5310.19595276
Record holderFounding benchmark
Solution methodHuman
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n19
CURRENT RECORD5805.858345926
Record holderFounding benchmark
Solution methodHuman
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n20
CURRENT RECORD6242.653092034
Record holderFounding benchmark
Solution methodHuman
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n21
CURRENT RECORD6561.344079972
Record holderFounding benchmark
Solution methodHuman
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n22
CURRENT RECORD7047.91006791
Record holderFounding benchmark
Solution methodHuman
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n23
CURRENT RECORD7577.420500292
Record holderFounding benchmark
Solution methodHuman
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n24
CURRENT RECORD8100.097599342
Record holderFounding benchmark
Solution methodHuman
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