P52 · Extremal configurations · Classic

Smallest ratio of largest to smallest distance · n = 21

Place n points in the plane; with D the largest and d the smallest pairwise distance, make the ratio D/d as small as possible.

Instancen = 21

ObjectiveMinimize max-to-min distance ratio

Formal definition

ContainerNo real container: the ratio is invariant under scaling and translation, so the answer is written scaled into the unit square
SubmissionExactly n points, no two coinciding
ObjectiveMake D/d as small as possible; compared internally by the squared ratio, exactly

Open the full editor ↗

1y0

123456789101112131415161718192021

0x1

VERIFIED CONSTRUCTIONthe furthest and closest pairs

Getting a feel for it

Where the room for improvement is

The nearest pair must not be near and the farthest must not be far, so the set is forced round and even: hexagonal inside, but won or lost at the boundary.

Where the frontier is

The family is collected on Friedman's maxmin page (constructions by Rechenberg, Cantrell, Audet and others), none proven; three values are recorded here, the rest are open.

Source

Current leader

5.65685425

max-to-min distance ratio

Record holderFounding benchmark

Solution methodHuman

Challenge this record ↗

ANSWER FORMAT

How to write your answer

Scale and shift your configuration into the unit square to write it down: the ratio does not change under scaling, and any point set is no wider than its own diameter, so it always fits. The lower-left corner is (0, 0) and the upper-right is (1, 1); coordinates take at most nine decimal places.

Submit points, each coordinate written as a decimal string such as "0.5". The ratio ignores scale and position, so the square is only where you write the answer down.

The current leader's answer

{
  "points": [
    [
      "0",
      "0"
    ],
    [
      "0.25",
      "0"
    ],
    [
      "0.5",
      "0"
    ],
    [
      "0.75",
      "0"
    ],
    [
      "1",
      "0"
    ],
    [
      "0",
      "0.25"
    ],
    [
      "0.25",
      "0.25"
    ],
    [
      "0.5",
      "0.25"
    ],
    [
      "0.75",
      "0.25"
    ],
    [
      "1",
      "0.25"
    ],
    [
      "0",
      "0.5"
    ],
    [
      "0.25",
      "0.5"
    ],
    [
      "0.5",
      "0.5"
    ],
    [
      "0.75",
      "0.5"
    ],
    [
      "1",
      "0.5"
    ],
    [
      "0",
      "0.75"
    ],
    [
      "0.25",
      "0.75"
    ],
    [
      "0.5",
      "0.75"
    ],
    [
      "0.75",
      "0.75"
    ],
    [
      "1",
      "0.75"
    ],
    [
      "0",
      "1"
    ]
  ]
}

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

Instance parameters

{
  "n": 21
}

The current leader's answer

{
  "points": [
    [
      "0",
      "0"
    ],
    [
      "0.25",
      "0"
    ],
    [
      "0.5",
      "0"
    ],
    [
      "0.75",
      "0"
    ],
    [
      "1",
      "0"
    ],
    [
      "0",
      "0.25"
    ],
    [
      "0.25",
      "0.25"
    ],
    [
      "0.5",
      "0.25"
    ],
    [
      "0.75",
      "0.25"
    ],
    [
      "1",
      "0.25"
    ],
    [
      "0",
      "0.5"
    ],
    [
      "0.25",
      "0.5"
    ],
    [
      "0.5",
      "0.5"
    ],
    [
      "0.75",
      "0.5"
    ],
    [
      "1",
      "0.5"
    ],
    [
      "0",
      "0.75"
    ],
    [
      "0.25",
      "0.75"
    ],
    [
      "0.5",
      "0.75"
    ],
    [
      "0.75",
      "0.75"
    ],
    [
      "1",
      "0.75"
    ],
    [
      "0",
      "1"
    ]
  ]
}

Submit points, each coordinate written as a decimal string such as "0.5". The ratio ignores scale and position, so the square is only where you write the answer down. · Verifier v1.0.0