P13 · Packing and covering · Classic

Packing circles of radius 1,2,…,n into a square · n = 3

Fit n circles of radii 1, 2, …, n, none overlapping, inside one square, making the side of that square as small as possible.

Instancen = 3

ObjectiveMinimize square side

Proven optimum8.535533906the two largest circles must be 5 apart, forcing side 5(1+√2/2)

FinishedAn answer reaching the ceiling is already published, so there is nothing left to take

Formal definition

ContainerA square of side side with the origin at its lower-left corner, where side is yours to choose — it is the score; the unit is the smallest circle
Submissionside and centers, the centres listed in order of radii 1, 2, …, n
ConstraintsCircle i has radius exactly i; no two overlap in their interiors; every circle lies wholly inside the square
ObjectiveMake the side of the square as small as possible

Open the editor and look ↗

8.535533906y0

123

0x8.535533906

VERIFIED CONSTRUCTIONside = 8.535533906

Getting a feel for it

Where the room for improvement is

The big circles set the skeleton and the small ones caulk the seams: each new largest circle can upend the whole previous layout.

Where the frontier is

n ≤ 4 follow from elementary centre-distance bounds (see the sub-problems). The circular-container version was the Zimmermann contest problem (see Packomania); this square version has no table, and the rest are open.

Source

Current leader

8.535533906

square side

OptimalThe optimum is proven; there is nothing left to gain

Record holderReference answer

Solution method—

Challenge this record ↗

ANSWER FORMAT

How to write your answer

The unit is the radius of the smallest circle: circle i has radius exactly i. The container is the square you name — lower-left corner at the origin (0, 0), upper-right at (side, side) — and a smaller side scores better. Coordinates and radii share one unit and are written as plain decimals such as "3.5", to at most nine decimal places.

Submit side and centers, listed in order of radius 1,2,…,n. Write every number as a decimal string, for example "3.5".

The current leader's answer

{
  "centers": [
    [
      "7.535533906",
      "1"
    ],
    [
      "6.535533906",
      "6.535533906"
    ],
    [
      "3",
      "3"
    ]
  ],
  "side": "8.535533906"
}

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

Instance parameters

{
  "n": 3
}

The current leader's answer

{
  "centers": [
    [
      "7.535533906",
      "1"
    ],
    [
      "6.535533906",
      "6.535533906"
    ],
    [
      "3",
      "3"
    ]
  ],
  "side": "8.535533906"
}

Submit side and centers, listed in order of radius 1,2,…,n. Write every number as a decimal string, for example "3.5". · Verifier v1.0.0