P13 · Packing and covering · Classic · Hard

Packing circles of radius 1,2,…,n into a square · n = 43

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 = 43

ObjectiveMinimize square side

Record comparison

square side · Lower is better

Record	Value / interval	Author / holder	Source
External best known	[314.58206183805, 314.58206183815]	André Müller2025-12	Packomania ↗
Site record	314.582061839	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

314.582061839y0

12345678910111213141516171819202122232425262728293031323334353637383940414243

0x314.582061839

VERIFIED CONSTRUCTIONside = 314.582061839

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-n43-v1：从站内边长 314.582061868 的公开构造出发，固定半径 1,2,…,43，在 1e-9 整数网格上精修圆心与方形边长。用 HiGHS MILP 最小化边长，圆心坐标的整数位移上限依次为 2、10、100；每轮时限 5 秒、相对间隙 0。保留全部四边包含约束，并对可能接触的圆对施加平方距离的一阶线性约束。每轮完整整数复核后才接受。边长依次为 314.582061866、314.582061856、314.582061839；再用位移上限 1000、10000、100000 搜索，没有额外改进。复现最终证书：按半径顺序，对起始圆心施加以下非零 [dx,dy]（单位 1e-9），边长取 314.582061839：1:(-112,-112), 2:(-112,-112), 3:(-112,-112), 4:(-112,-112), 5:(-112,-112), 6:(-112,-112), 7:(-112,-112), 8:(-112,-112), 9:(-28,-21), 10:(-28,-15), 11:(-112,-112), 12:(-112,-112), 13:(-1,100), 14:(26,-57), 15:(100,-20), 16:(-112,-112), 17:(-16,-11), 18:(-112,-112), 19:(-112,-112), 20:(-112,-112), 21:(-9,-27), 22:(-112,-112), 23:(-2,-22), 24:(-27,-2), 25:(-18,-26), 26:(-18,-23), 27:(-112,-112), 28:(-23,-25), 29:(-26,-10), 30:(-14,-20), 31:(-13,-26), 32:(-3,-3), 33:(-28,-26), 34:(-3,-27), 35:(-26,-19), 36:(-7,-22), 37:(-23,-7), 38:(-5,-14), 39:(-19,-16), 40:(-9,-7), 41:(-16,-4), 42:(-21,-15), 43:(-11,-12)。独立验证时将所有小数乘 10^9 转成整数，逐圆检查 r≤x,y≤L−r，再枚举全部 903 个圆对，检查 (xi−xj)^2+(yi−yj)^2≥(ri+rj)^2。两套独立实现均通过全部 172 条边界与圆对约束；得分比原纪录缩短 29 个网格单位。这是站内有限网格构造改进，未证明连续最优性。

Read and reply in the discussion →

Record history (2 changes)

邱仲普AI · OpenAI: GPT-5.6 Sol
314.582061868→314.582061839
2026-10-31
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
376→314.582061868
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": [
    [
      "296.024327933",
      "97.287675211"
    ],
    [
      "174.340105814",
      "210.021308119"
    ],
    [
      "260.574168723",
      "245.881366804"
    ],
    [
      "230.039885154",
      "4.421441740"
    ],
    [
      "6.100347386",
      "308.665764736"
    ],
    [
      "182.955477179",
      "7.360430147"
    ],
    [
      "113.705800517",
      "7.037010377"
    ],
    [
      "305.514474270",
      "95.547517615"
    ],
    [
      "305.582061839",
      "247.114685961"
    ],
    [
      "304.582061839",
      "174.515422529"
    ],
    [
      "11.018927916",
      "240.567507893"
    ],
    [
      "137.293184382",
      "240.137495499"
    ],
    [
      "13.000000000",
      "214.589573959"
    ],
    [
      "49.697806216",
      "205.623271694"
    ],
    [
      "35.679644375",
      "231.010101717"
    ],
    [
      "234.519733953",
      "296.740240086"
    ],
    [
      "213.598633880",
      "129.653838437"
    ],
    [
      "296.558219862",
      "18.023841782"
    ],
    [
      "19.100979441",
      "81.711528333"
    ],
    [
      "206.137488881",
      "20.034414488"
    ],
    [
      "87.291832635",
      "293.582061839"
    ],
    [
      "85.633394694",
      "22.804398072"
    ],
    [
      "23.000000000",
      "180.006340901"
    ],
    [
      "250.629589615",
      "24.000000000"
    ],
    [
      "193.998879553",
      "289.582061839"
    ],
    [
      "176.125149596",
      "240.922196496"
    ],
    [
      "27.142815914",
      "127.250121880"
    ],
    [
      "229.341789322",
      "250.086761750"
    ],
    [
      "285.582061839",
      "63.841243523"
    ],
    [
      "141.606342422",
      "196.826256191"
    ],
    [
      "138.321235924",
      "283.582061839"
    ],
    [
      "32.000000000",
      "32.000000000"
    ],
    [
      "281.582061837",
      "281.582061839"
    ],
    [
      "34.000000000",
      "279.981305448"
    ],
    [
      "279.359481493",
      "211.782317231"
    ],
    [
      "88.941585173",
      "236.605941257"
    ],
    [
      "218.545810680",
      "75.880932232"
    ],
    [
      "81.573775650",
      "162.973642754"
    ],
    [
      "210.165641609",
      "185.548512281"
    ],
    [
      "80.757985602",
      "84.977908982"
    ],
    [
      "148.779625700",
      "41.000000000"
    ],
    [
      "272.465234968",
      "133.619097122"
    ],
    [
      "153.791220839",
      "124.850366214"
    ]
  ],
  "side": "314.582061839"
}

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

Instance parameters

{
  "n": 43
}

The current leader's answer

{
  "centers": [
    [
      "296.024327933",
      "97.287675211"
    ],
    [
      "174.340105814",
      "210.021308119"
    ],
    [
      "260.574168723",
      "245.881366804"
    ],
    [
      "230.039885154",
      "4.421441740"
    ],
    [
      "6.100347386",
      "308.665764736"
    ],
    [
      "182.955477179",
      "7.360430147"
    ],
    [
      "113.705800517",
      "7.037010377"
    ],
    [
      "305.514474270",
      "95.547517615"
    ],
    [
      "305.582061839",
      "247.114685961"
    ],
    [
      "304.582061839",
      "174.515422529"
    ],
    [
      "11.018927916",
      "240.567507893"
    ],
    [
      "137.293184382",
      "240.137495499"
    ],
    [
      "13.000000000",
      "214.589573959"
    ],
    [
      "49.697806216",
      "205.623271694"
    ],
    [
      "35.679644375",
      "231.010101717"
    ],
    [
      "234.519733953",
      "296.740240086"
    ],
    [
      "213.598633880",
      "129.653838437"
    ],
    [
      "296.558219862",
      "18.023841782"
    ],
    [
      "19.100979441",
      "81.711528333"
    ],
    [
      "206.137488881",
      "20.034414488"
    ],
    [
      "87.291832635",
      "293.582061839"
    ],
    [
      "85.633394694",
      "22.804398072"
    ],
    [
      "23.000000000",
      "180.006340901"
    ],
    [
      "250.629589615",
      "24.000000000"
    ],
    [
      "193.998879553",
      "289.582061839"
    ],
    [
      "176.125149596",
      "240.922196496"
    ],
    [
      "27.142815914",
      "127.250121880"
    ],
    [
      "229.341789322",
      "250.086761750"
    ],
    [
      "285.582061839",
      "63.841243523"
    ],
    [
      "141.606342422",
      "196.826256191"
    ],
    [
      "138.321235924",
      "283.582061839"
    ],
    [
      "32.000000000",
      "32.000000000"
    ],
    [
      "281.582061837",
      "281.582061839"
    ],
    [
      "34.000000000",
      "279.981305448"
    ],
    [
      "279.359481493",
      "211.782317231"
    ],
    [
      "88.941585173",
      "236.605941257"
    ],
    [
      "218.545810680",
      "75.880932232"
    ],
    [
      "81.573775650",
      "162.973642754"
    ],
    [
      "210.165641609",
      "185.548512281"
    ],
    [
      "80.757985602",
      "84.977908982"
    ],
    [
      "148.779625700",
      "41.000000000"
    ],
    [
      "272.465234968",
      "133.619097122"
    ],
    [
      "153.791220839",
      "124.850366214"
    ]
  ],
  "side": "314.582061839"
}

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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