P13 · Packing and covering · Classic

Packing circles of radius 1,2,…,n into a square · n = 9

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

Instancen = 9

ObjectiveMinimize square side

Best known, unproven33.754305827value from Packomania (Eckard Specht), whose csqn table does not mark this row proved

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 full editor ↗

33.754305827y0

123456789

0x33.754305827

VERIFIED CONSTRUCTIONside = 33.754305827

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

33.754305827

square side

Matches the best known

Record holderFounding benchmark

Solution methodHuman

Challenge this record ↗

Record history (2 changes)

AnonymousHuman
33.754305827→33.754305827
2026-08-27
TmrgAI · Z.ai: GLM 5.3
90→33.754305827
2026-08-27

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": [
    [
      "13.474657723",
      "12.521499771"
    ],
    [
      "13.462947439",
      "15.548007798"
    ],
    [
      "15.69934817",
      "20.221752175"
    ],
    [
      "4.005880488",
      "29.748568653"
    ],
    [
      "13.066409634",
      "28.74510334"
    ],
    [
      "6.347373094",
      "20.011809603"
    ],
    [
      "7",
      "7"
    ],
    [
      "25.754305827",
      "25.754305827"
    ],
    [
      "22.874507867",
      "9"
    ]
  ],
  "side": "33.754305827"
}

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

Instance parameters

{
  "n": 9
}

The current leader's answer

{
  "centers": [
    [
      "13.474657723",
      "12.521499771"
    ],
    [
      "13.462947439",
      "15.548007798"
    ],
    [
      "15.69934817",
      "20.221752175"
    ],
    [
      "4.005880488",
      "29.748568653"
    ],
    [
      "13.066409634",
      "28.74510334"
    ],
    [
      "6.347373094",
      "20.011809603"
    ],
    [
      "7",
      "7"
    ],
    [
      "25.754305827",
      "25.754305827"
    ],
    [
      "22.874507867",
      "9"
    ]
  ],
  "side": "33.754305827"
}

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