P33 · Extremal configurations · Classic · Weak baseline · Hard

Riesz 2-energy in a square · n = 8

Also known asRiesz energyRiesz s-energyminimum energy point configuration

Place n points in the unit square so as to minimize the sum of 1/distance² over all pairs.

Instancen = 8

ObjectiveMinimize Riesz 2-energy

Record comparison

Riesz 2-energy · Lower is better

Record	Value / interval	Author / holder	Source
External best known	53.4	See source attribution	Proof / source four corners plus four midpoints, E = 53.4
Site record	No player record yet	—	—

Formal definition

ContainerThe unit square of side 1, with lower-left corner at (0, 0) and upper-right corner at (1, 1)
SubmissionExactly n distinct points
ConstraintsEvery point lies inside the container or on its boundary
EnergyE₂(P) is the Riesz 2-energy of P $E_{2} (P) = \sum_{1 \leq i < j \leq n} \frac{1}{{‖ p_{i} - p_{j} ‖}^{2}}$
ScoreThe verifier rounds each pair's energy upward to the 10⁻⁹ grid before summing, and displays score(P)/10⁹ $score (P) = \sum_{1 \leq i < j \leq n} ⌈ \frac{10^{9}}{{‖ p_{i} - p_{j} ‖}^{2}} ⌉$
ObjectiveMinimize score(P), which approximates E₂(P) with an upward rounding error of at most 10⁻⁹ per pair

Open full editor ↗

Layout shown: starter layout

1y0

12345678

0x1

VERIFIED CONSTRUCTIONthe closest pair

Getting a feel for it

Structural features

Because 1/distance² strongly repels nearby points, small point sets are pushed entirely onto the boundary; as n grows, points organize into concentric interior shells. The number of shells and the population of each shell change abruptly at specific values of n.

Known results

For the general theory of pairwise 1/r² energy, including asymptotics and point separation, see Borodachov, Hardin, and Saff, Discrete Energy on Rectifiable Sets (2019). No published table of per-n optima exists for the unit square, and every instance here is open.

Source

LAYOUT & HISTORY

Starter layout

Inspect the layout, solver notes and record history.

Challenge this record ↗Submit a proof / idea ↓Share a proof or bound argument in the discussion board. Accepted contributions earn proof points.

Record history (2 changes)

Reference answerHuman
53.4→53.4
2026-08-26
Yifan YaoHuman
53.4→53.4
2026-08-24

ANSWER FORMAT

How to write your answer

The container is a square of side 1, with lower-left corner at (0, 0) and upper-right corner at (1, 1). Write coordinates as decimal strings such as "0.5", with at most nine decimal places.

Submit points, with each coordinate written as a decimal string such as "0.5". The score is the sum of 1/distance² over all pairs; smaller is better.

Starter answer

{
  "points": [
    [
      "0",
      "0"
    ],
    [
      "1",
      "0"
    ],
    [
      "1",
      "1"
    ],
    [
      "0",
      "1"
    ],
    [
      "0.5",
      "0"
    ],
    [
      "1",
      "0.5"
    ],
    [
      "0.5",
      "1"
    ],
    [
      "0",
      "0.5"
    ]
  ]
}

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

Instance parameters

{
  "n": 8
}

Starter answer

{
  "points": [
    [
      "0",
      "0"
    ],
    [
      "1",
      "0"
    ],
    [
      "1",
      "1"
    ],
    [
      "0",
      "1"
    ],
    [
      "0.5",
      "0"
    ],
    [
      "1",
      "0.5"
    ],
    [
      "0.5",
      "1"
    ],
    [
      "0",
      "0.5"
    ]
  ]
}

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

DISCUSSION

Discussion

Discuss strategies, share construction methods, or ask questions. Any signed-in user can post under their public byline and unique #UID. New posts appear after moderation.

直到现在看不懂NUE_13#310 replies30d ago