P34 · Extremal configurations · Classic · Weak baseline

Riesz 2-energy in a disc · n = 9

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

Instancen = 9

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

Open the full editor ↗

2y0

123456789

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

29.00000004

Riesz 2-energy

Record holderlird

Solution methodHuman

Challenge this record ↗

Record history (1 changes)

lirdHuman
696.600005604→29.00000004
2026-08-26

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": [
    [
      "1.980785280",
      "1.195090322"
    ],
    [
      "1.555570233",
      "1.831469612"
    ],
    [
      "0.804909678",
      "1.980785280"
    ],
    [
      "0.168530388",
      "1.555570233"
    ],
    [
      "0.019214720",
      "0.804909678"
    ],
    [
      "0.444429767",
      "0.168530388"
    ],
    [
      "1.195090322",
      "0.019214720"
    ],
    [
      "1.831469612",
      "0.444429767"
    ],
    [
      "1.000000000",
      "1.000000000"
    ]
  ]
}

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

Instance parameters

{
  "n": 9
}

The current leader's answer

{
  "points": [
    [
      "1.980785280",
      "1.195090322"
    ],
    [
      "1.555570233",
      "1.831469612"
    ],
    [
      "0.804909678",
      "1.980785280"
    ],
    [
      "0.168530388",
      "1.555570233"
    ],
    [
      "0.019214720",
      "0.804909678"
    ],
    [
      "0.444429767",
      "0.168530388"
    ],
    [
      "1.195090322",
      "0.019214720"
    ],
    [
      "1.831469612",
      "0.444429767"
    ],
    [
      "1.000000000",
      "1.000000000"
    ]
  ]
}

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