P13 · Packing and covering · Classic · Hard

Packing circles of radius 1,2,…,n into a square · n = 21

Also known aspacking circles with radii 1 to n in a squaregraduated circle packing

Pack n non-overlapping circles of radii 1, 2, …, n inside an enclosing square to minimize its side length.

Instancen = 21

ObjectiveMinimize square side

Record comparison

square side · Lower is better

Record	Value / interval	Author / holder	Source
External best known	[110.56029719535, 110.56029719545]	Kun He, Menglong Huang, Chenkai Yang2015-04-19	Packomania ↗
Site record	110.560297197	Stonebubu	Record details ↓

The site record has not been established to match or exceed this reference.

Formal definition

ContainerA square [0, side] × [0, side] with the origin at its lower-left corner, where the smallest circle has unit radius 1 and side is the objective value to minimize
SubmissionSquare side and centers, listed in order of radii 1, 2, …, n
ConstraintsCircle i has radius i; no two interiors overlap; every circle lies wholly inside the square container
ObjectiveMinimize the side length of the enclosing square

Open full editor ↗

Layout shown: site record

110.560297197y0

123456789101112131415161718192021

0x110.560297197

VERIFIED CONSTRUCTIONside = 110.560297197

Getting a feel for it

Corner placement and interstitial gaps

Because a square has four right-angle corners, seating the largest few circles into the corners and staggering them across the diagonals is key to minimizing the side length, while smaller circles fit into the triangular pockets left behind. Increasing n by 1 introduces a new largest circle that often forces a reassignment of the four corners.

Proven optima and open frontier

Optimality for n ≤ 4 follows from elementary center-distance projections among the largest circles (see the instance notes). Packomania's csqn table lists best known values up to n = 100, and its ratio column is this problem's side. The table marks none of them proved, so n ≥ 5 remains open. The circular-container counterpart was a Zimmermann contest problem, also on Packomania.

Source

LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holderStonebubu

Solution methodAI · Z.ai: GLM 5.3 Flash

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

Record holder's solver noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

Record history (1 changes)

StonebubuAI · Z.ai: GLM 5.3 Flash
110.560297198→110.560297197
2026-09-02

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 the square side length and the circle centers listed in order of radii 1, 2, …, n. Write each number as a decimal string such as "3.5".

The current leader's answer

{
  "centers": [
    [
      "6.288438811",
      "1.113913211"
    ],
    [
      "40.518642606",
      "20.429448833"
    ],
    [
      "35.339064893",
      "3.055240234"
    ],
    [
      "106.511721270",
      "27.755219860"
    ],
    [
      "5.016297609",
      "38.611188890"
    ],
    [
      "76.885270951",
      "85.955954704"
    ],
    [
      "34.826520073",
      "81.348100483"
    ],
    [
      "84.738434853",
      "74.322192044"
    ],
    [
      "101.557045288",
      "71.789441562"
    ],
    [
      "46.792774643",
      "10.039273346"
    ],
    [
      "66.691063509",
      "99.560297197"
    ],
    [
      "43.712812922",
      "98.560297197"
    ],
    [
      "97.560297197",
      "13.000000000"
    ],
    [
      "70.578822070",
      "14.000000000"
    ],
    [
      "95.560297197",
      "95.560297197"
    ],
    [
      "16.000000000",
      "94.560297197"
    ],
    [
      "50.112833768",
      "37.283971372"
    ],
    [
      "57.776931735",
      "71.434543877"
    ],
    [
      "19.009616584",
      "19.009754041"
    ],
    [
      "20.000000000",
      "58.783209556"
    ],
    [
      "87.238245280",
      "45.414785818"
    ]
  ],
  "side": "110.560297197"
}

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

Instance parameters

{
  "n": 21
}

The current leader's answer

{
  "centers": [
    [
      "6.288438811",
      "1.113913211"
    ],
    [
      "40.518642606",
      "20.429448833"
    ],
    [
      "35.339064893",
      "3.055240234"
    ],
    [
      "106.511721270",
      "27.755219860"
    ],
    [
      "5.016297609",
      "38.611188890"
    ],
    [
      "76.885270951",
      "85.955954704"
    ],
    [
      "34.826520073",
      "81.348100483"
    ],
    [
      "84.738434853",
      "74.322192044"
    ],
    [
      "101.557045288",
      "71.789441562"
    ],
    [
      "46.792774643",
      "10.039273346"
    ],
    [
      "66.691063509",
      "99.560297197"
    ],
    [
      "43.712812922",
      "98.560297197"
    ],
    [
      "97.560297197",
      "13.000000000"
    ],
    [
      "70.578822070",
      "14.000000000"
    ],
    [
      "95.560297197",
      "95.560297197"
    ],
    [
      "16.000000000",
      "94.560297197"
    ],
    [
      "50.112833768",
      "37.283971372"
    ],
    [
      "57.776931735",
      "71.434543877"
    ],
    [
      "19.009616584",
      "19.009754041"
    ],
    [
      "20.000000000",
      "58.783209556"
    ],
    [
      "87.238245280",
      "45.414785818"
    ]
  ],
  "side": "110.560297197"
}

Submit the square side length and the circle centers listed in order of radii 1, 2, …, n. Write each number as a decimal string such as "3.5". · Verifier v1.0.0

DISCUSSION

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