P34 · Extremal configurations · Classic · Weak baseline

Riesz 2-energy in a disc · n = 20

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

Instancen = 20

ObjectiveMinimize Riesz 2-energy

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

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2y0

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0x2

VERIFIED CONSTRUCTIONthe closest pair

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

Current leader

6242.653092034

Riesz 2-energy

Record holderFounding benchmark

Solution methodHuman

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ANSWER FORMAT

How to write your answer

The container is a circle of radius 1 centred at (1, 1), so coordinates run from 0 to 2. Coordinates are written as plain decimals such as "0.5", to at most nine decimal places.

Submit points, each coordinate written as a decimal string such as "0.5". The score is the sum of 1/distance² over every pair, and smaller is better.

The current leader's answer

{
  "points": [
    [
      "0.8",
      "0.8"
    ],
    [
      "0.9",
      "0.8"
    ],
    [
      "1",
      "0.8"
    ],
    [
      "1.1",
      "0.8"
    ],
    [
      "1.2",
      "0.8"
    ],
    [
      "0.8",
      "0.9"
    ],
    [
      "0.9",
      "0.9"
    ],
    [
      "1",
      "0.9"
    ],
    [
      "1.1",
      "0.9"
    ],
    [
      "1.2",
      "0.9"
    ],
    [
      "0.8",
      "1"
    ],
    [
      "0.9",
      "1"
    ],
    [
      "1",
      "1"
    ],
    [
      "1.1",
      "1"
    ],
    [
      "1.2",
      "1"
    ],
    [
      "0.8",
      "1.1"
    ],
    [
      "0.9",
      "1.1"
    ],
    [
      "1",
      "1.1"
    ],
    [
      "1.1",
      "1.1"
    ],
    [
      "1.2",
      "1.1"
    ]
  ]
}

Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 20
}

The current leader's answer

{
  "points": [
    [
      "0.8",
      "0.8"
    ],
    [
      "0.9",
      "0.8"
    ],
    [
      "1",
      "0.8"
    ],
    [
      "1.1",
      "0.8"
    ],
    [
      "1.2",
      "0.8"
    ],
    [
      "0.8",
      "0.9"
    ],
    [
      "0.9",
      "0.9"
    ],
    [
      "1",
      "0.9"
    ],
    [
      "1.1",
      "0.9"
    ],
    [
      "1.2",
      "0.9"
    ],
    [
      "0.8",
      "1"
    ],
    [
      "0.9",
      "1"
    ],
    [
      "1",
      "1"
    ],
    [
      "1.1",
      "1"
    ],
    [
      "1.2",
      "1"
    ],
    [
      "0.8",
      "1.1"
    ],
    [
      "0.9",
      "1.1"
    ],
    [
      "1",
      "1.1"
    ],
    [
      "1.1",
      "1.1"
    ],
    [
      "1.2",
      "1.1"
    ]
  ]
}

Submit points, each coordinate written as a decimal string such as "0.5". The score is the sum of 1/distance² over every pair, and smaller is better. · Verifier v1.0.0