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P11 · Packing and covering · Classic

Equal-circle packing in a right triangle · n = 11

Place n non-overlapping equal circles inside the right triangle with legs 1 and 0.75, making the common radius as large as possible.

Instancen = 11
ObjectiveMaximize common radius

Formal definition

  • ContainerA right triangle: the right angle at the origin (0, 0), one leg along the x-axis to (1, 0), the other along the y-axis to (0, 0.75)
  • SubmissionExactly n circles: one shared radius and n centres
  • ConstraintsEvery circle lies wholly inside the container; no two overlap in their interiors, tangency allowed
  • ObjectiveMake the common radius as large as possible
Open the full editor ↗
0.75y0
0x1
VERIFIED CONSTRUCTIONr = 0.041666666

Getting a feel for it

Where the room for improvement is

Optimal packings are jammed contact structures: circles brace against each other and the boundary, with tilted rows, offsets, and the odd rattler touching nothing. Neat grids are almost never optimal.

Where the frontier is

This particular triangle, legs 1 and 0.75, has no literature table; close relatives such as the isosceles right triangle are collected on Friedman's Packing Center. No values are recorded here, and every n is open.

Source
Current leader

0.041666666

common radius

Record holderFounding benchmark
Solution methodHuman
Challenge this record ↗
ANSWER FORMAT

How to write your answer

The container is a right triangle: the right angle is the origin (0, 0), one leg runs along the x axis to (1, 0), the other along the y axis to (0, 0.75), and the hypotenuse joins those two points. Coordinates and lengths share one unit and are written as plain decimals such as "0.15", to at most nine decimal places.

Submit radius and centers. Write every number as a decimal string, for example "0.15".

The current leader's answer

{
  "centers": [
    [
      "0.041666666",
      "0.041666666"
    ],
    [
      "0.124999998",
      "0.041666666"
    ],
    [
      "0.20833333",
      "0.041666666"
    ],
    [
      "0.291666662",
      "0.041666666"
    ],
    [
      "0.374999994",
      "0.041666666"
    ],
    [
      "0.458333326",
      "0.041666666"
    ],
    [
      "0.541666658",
      "0.041666666"
    ],
    [
      "0.62499999",
      "0.041666666"
    ],
    [
      "0.708333322",
      "0.041666666"
    ],
    [
      "0.791666654",
      "0.041666666"
    ],
    [
      "0.874999986",
      "0.041666666"
    ]
  ],
  "radius": "0.041666666"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 11
}

The current leader's answer

{
  "centers": [
    [
      "0.041666666",
      "0.041666666"
    ],
    [
      "0.124999998",
      "0.041666666"
    ],
    [
      "0.20833333",
      "0.041666666"
    ],
    [
      "0.291666662",
      "0.041666666"
    ],
    [
      "0.374999994",
      "0.041666666"
    ],
    [
      "0.458333326",
      "0.041666666"
    ],
    [
      "0.541666658",
      "0.041666666"
    ],
    [
      "0.62499999",
      "0.041666666"
    ],
    [
      "0.708333322",
      "0.041666666"
    ],
    [
      "0.791666654",
      "0.041666666"
    ],
    [
      "0.874999986",
      "0.041666666"
    ]
  ],
  "radius": "0.041666666"
}

Submit radius and centers. Write every number as a decimal string, for example "0.15". · Verifier v1.0.0