P33 · Extremal configurations · Classic · Weak baseline

Riesz 2-energy in a square · n = 15

Place n points inside the unit square, minimizing the sum of 1/distance² taken over every pair.

Instancen = 15

ObjectiveMinimize Riesz 2-energy

Formal definition

ContainerThe container is a square of side 1, with the origin (0, 0) at its lower-left corner and (1, 1) at its upper-right.
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 ↗

1y0

123456789101112131415

0x1

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

The general theory of pairwise 1/r² energy (asymptotics, separation) is Borodachov, Hardin and Saff, Discrete Energy on Rectifiable Sets (2019); a per-n table of optima in the square does not exist, and every n here is open. The few trivial closed forms are proved on site.

Source

Current leader

355.861648888

Riesz 2-energy

Record holderlird

Solution methodHuman

Challenge this record ↗

Record history (1 changes)

lirdHuman
10613.442735051→355.861648888
2026-08-26

ANSWER FORMAT

How to write your answer

The container is a square of side 1, with the origin (0, 0) at its lower-left corner and (1, 1) at its upper-right. 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": [
    [
      "0.000000000",
      "1.000000000"
    ],
    [
      "0.500000000",
      "1.000000000"
    ],
    [
      "1.000000000",
      "1.000000000"
    ],
    [
      "1.000000000",
      "0.659666666"
    ],
    [
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.500000000",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.662666666"
    ],
    [
      "0.246000000",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.340333333"
    ],
    [
      "0.246000000",
      "1.000000000"
    ],
    [
      "0.500000000",
      "0.500000000"
    ],
    [
      "0.754000000",
      "1.000000000"
    ],
    [
      "1.000000000",
      "0.337333333"
    ],
    [
      "0.754000000",
      "0.000000000"
    ]
  ]
}

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

Instance parameters

{
  "n": 15
}

The current leader's answer

{
  "points": [
    [
      "0.000000000",
      "1.000000000"
    ],
    [
      "0.500000000",
      "1.000000000"
    ],
    [
      "1.000000000",
      "1.000000000"
    ],
    [
      "1.000000000",
      "0.659666666"
    ],
    [
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.500000000",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.662666666"
    ],
    [
      "0.246000000",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.340333333"
    ],
    [
      "0.246000000",
      "1.000000000"
    ],
    [
      "0.500000000",
      "0.500000000"
    ],
    [
      "0.754000000",
      "1.000000000"
    ],
    [
      "1.000000000",
      "0.337333333"
    ],
    [
      "0.754000000",
      "0.000000000"
    ]
  ]
}

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