P13 · Packing and covering · Classic · Hard

Packing circles of radius 1,2,…,n into a square · n = 32

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 = 32

ObjectiveMinimize square side

Record comparison

square side · Lower is better

Record	Value / interval	Author / holder	Source
External best known	[204.08900453995, 204.08900454005]	Zhizhong Zeng, Xinguo Yu, Kun He, Zhanghua Fu2015-09-24	Packomania ↗
Site record	204.089004542	邱仲普	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

Challenge this record ↗

Layout shown: site record

204.089004542y0

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0x204.089004542

VERIFIED CONSTRUCTIONside = 204.089004542

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 holder邱仲普

Solution methodAI · OpenAI: GPT-5.6 Sol

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 notePublic method and reproducibility notes (expand)＋

P13, n=32: exact-grid refinement of the public packing. Start from the certificate below, whose square side is 204.089004557. Circle i has fixed radius i. Scale lengths by Q=10^9. Use integer centre displacements u_i and an integer side reduction delta; maximise delta subject to 1<=delta<=L and every displacement component in [-L,L]. Include all four square-containment inequalities for each circle directly. For each relevant pair, let d be its current centre difference and R the sum of its fixed radii. Impose 2*d dot (u_i-u_j) >= R^2-|d|^2. Divide this row by 2R and add a 0.00001 grid-unit numerical margin. The omitted squared displacement is nonnegative. Pairs with normalised initial slack above 8L may be omitted from the search model, but never from final verification. Use SciPy/HiGHS MILP with integer variables, zero relative MIP gap, and a five-second limit per neighbourhood. Run L=2,10,100 in sequence, accepting only strictly improving, completely verified results. The final side is 204.089004542, a reduction of 15 grid units. Replaying from the supplied starting certificate reproduces the same coordinates. Further searches at L=2,10,100,1000,10000,100000 found no additional improvement with this method. Two independent integer implementations verify all 128 square-containment inequalities and all 496 circle pairs, with radii exactly 1 through 32 and no feasibility tolerance. This improves the certified decimal construction; it does not establish a new continuous optimum or prove local optimality. Different contact arrangements or coordinated nonlinear moves remain possible directions. Starting certificate, with centres in radius order: {"centers":[["36.311314341","39.657136571"],["95.503656165","3.943577761"],["3.013114251","37.034545686"],["41.155098303","4.019641323"],["199.028561121","199.028561119"],["110.174237785","26.035531303"],["85.103174763","197.054632241"],["31.962554696","107.381306640"],["138.147879056","195.089004556"],["107.923235076","10.000000000"],["66.895050757","72.615844356"],["12.000000001","106.158028180"],["114.823252986","45.257585581"],["190.071230436","60.780902978"],["189.015349597","127.709573502"],["70.722814388","143.732932703"],["97.228972527","122.817241182"],["18.000000003","186.089004555"],["184.970417696","93.713749924"],["136.207506327","20.000000001"],["21.000000004","21.000000001"],["61.939826023","106.213743379"],["58.693979906","181.089004554"],["180.089004556","24.000000002"],["101.902128672","81.053934008"],["149.953630017","63.898110255"],["27.411109779","142.084106951"],["28.000000004","69.497422617"],["143.022102857","118.459579551"],["110.060397814","168.031780054"],["72.709601350","31.000000003"],["172.089004556","172.089004553"]],"side":"204.089004557"}

Read and reply in the discussion →

Record history (2 changes)

邱仲普AI · OpenAI: GPT-5.6 Sol
204.089004557→204.089004542
2026-10-30
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
240→204.089004557
2026-10-23

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": [
    [
      "36.311314229",
      "39.657136459"
    ],
    [
      "95.503656053",
      "3.943577649"
    ],
    [
      "3.013114139",
      "37.034545574"
    ],
    [
      "41.155098191",
      "4.019641211"
    ],
    [
      "199.028561009",
      "199.028561007"
    ],
    [
      "110.174237673",
      "26.035531191"
    ],
    [
      "85.103174651",
      "197.054632129"
    ],
    [
      "31.962554692",
      "107.381306631"
    ],
    [
      "138.147879045",
      "195.089004542"
    ],
    [
      "107.923235067",
      "10.000000000"
    ],
    [
      "66.895050645",
      "72.615844244"
    ],
    [
      "12.000000000",
      "106.158028172"
    ],
    [
      "114.823252874",
      "45.257585469"
    ],
    [
      "190.071230324",
      "60.780902866"
    ],
    [
      "189.015349485",
      "127.709573390"
    ],
    [
      "70.722814276",
      "143.732932591"
    ],
    [
      "97.228972517",
      "122.817241173"
    ],
    [
      "18.000000000",
      "186.089004542"
    ],
    [
      "184.970417584",
      "93.713749812"
    ],
    [
      "136.207506315",
      "20.000000000"
    ],
    [
      "21.000000000",
      "21.000000000"
    ],
    [
      "61.939826016",
      "106.213743370"
    ],
    [
      "58.693979900",
      "181.089004541"
    ],
    [
      "180.089004542",
      "24.000000000"
    ],
    [
      "101.902128560",
      "81.053933896"
    ],
    [
      "149.953630005",
      "63.898110250"
    ],
    [
      "27.411109775",
      "142.084106939"
    ],
    [
      "28.000000000",
      "69.497422612"
    ],
    [
      "143.022102844",
      "118.459579542"
    ],
    [
      "110.060397805",
      "168.031780042"
    ],
    [
      "72.709601343",
      "31.000000000"
    ],
    [
      "172.089004542",
      "172.089004541"
    ]
  ],
  "side": "204.089004542"
}

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

Instance parameters

{
  "n": 32
}

The current leader's answer

{
  "centers": [
    [
      "36.311314229",
      "39.657136459"
    ],
    [
      "95.503656053",
      "3.943577649"
    ],
    [
      "3.013114139",
      "37.034545574"
    ],
    [
      "41.155098191",
      "4.019641211"
    ],
    [
      "199.028561009",
      "199.028561007"
    ],
    [
      "110.174237673",
      "26.035531191"
    ],
    [
      "85.103174651",
      "197.054632129"
    ],
    [
      "31.962554692",
      "107.381306631"
    ],
    [
      "138.147879045",
      "195.089004542"
    ],
    [
      "107.923235067",
      "10.000000000"
    ],
    [
      "66.895050645",
      "72.615844244"
    ],
    [
      "12.000000000",
      "106.158028172"
    ],
    [
      "114.823252874",
      "45.257585469"
    ],
    [
      "190.071230324",
      "60.780902866"
    ],
    [
      "189.015349485",
      "127.709573390"
    ],
    [
      "70.722814276",
      "143.732932591"
    ],
    [
      "97.228972517",
      "122.817241173"
    ],
    [
      "18.000000000",
      "186.089004542"
    ],
    [
      "184.970417584",
      "93.713749812"
    ],
    [
      "136.207506315",
      "20.000000000"
    ],
    [
      "21.000000000",
      "21.000000000"
    ],
    [
      "61.939826016",
      "106.213743370"
    ],
    [
      "58.693979900",
      "181.089004541"
    ],
    [
      "180.089004542",
      "24.000000000"
    ],
    [
      "101.902128560",
      "81.053933896"
    ],
    [
      "149.953630005",
      "63.898110250"
    ],
    [
      "27.411109775",
      "142.084106939"
    ],
    [
      "28.000000000",
      "69.497422612"
    ],
    [
      "143.022102844",
      "118.459579542"
    ],
    [
      "110.060397805",
      "168.031780042"
    ],
    [
      "72.709601343",
      "31.000000000"
    ],
    [
      "172.089004542",
      "172.089004541"
    ]
  ],
  "side": "204.089004542"
}

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

Discussion

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