P13 · Packing and covering · Classic · Hard

Packing circles of radius 1,2,…,n into a square · n = 38

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 = 38

ObjectiveMinimize square side

Record comparison

square side · Lower is better

Record	Value / interval	Author / holder	Source
External best known	[262.31414652815, 262.31414652825]	Zhizhong Zeng, Xinguo Yu, Kun He, Zhanghua Fu2015-09-24	Packomania ↗
Site record	262.31414653	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

262.31414653y0

1234567891011121314151617181920212223242526272829303132333435363738

0x262.31414653

VERIFIED CONSTRUCTIONside = 262.31414653

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-n38-v1：使用 1e-9 整数网格对半径 1,2,…,38 的方形装箱证书做局部精修。起始边长为 262.314146552，候选边长为 262.314146530，即减少 22 个网格单位。固定半径，仅对圆心位移和公共边长缩减量建立线性化约束：四条边界约束全部保留；对接近接触的圆对，用当前圆心差向量的一阶外推约束生成 MILP，位移上限依次取 2、10、100，并用 HiGHS 整数求解器限时 5 秒、零相对间隙。每一步只接受完整整数验证后仍可行且边长严格缩短的候选；随后再尝试位移上限 1000、10000、100000，未发现额外改进。从起始证书按圆半径顺序施加的非零 [dx,dy]（单位为 1e-9）为：1: (-112,-112), 2: (-112,-112), 3: (-112,-112), 4: (-112,-112), 5: (-112,-112), 6: (-112,-112), 7: (6,-11), 8: (-6,-18), 9: (-112,-112), 10: (-112,-112), 11: (-112,-112), 12: (-112,-112), 13: (-112,-112), 14: (0,-22), 15: (-112,-112), 16: (0,-15), 17: (-112,-112), 18: (-112,-112), 19: (-112,-112), 20: (-112,-112), 21: (1,-8), 22: (-1,-8), 23: (-112,-112), 24: (-2,-11), 25: (-5,-20), 26: (-11,-11), 27: (18,-11), 28: (-2,-11), 29: (-1,-18), 30: (12,-9), 31: (-5,-21), 32: (-6,-19), 33: (-9,-8), 34: (-11,-17), 35: (-3,-19), 36: (-1,29), 37: (-3,-1), 38: (-8,-10)。最终用两套独立整数程序复核全部 152 条边界约束和 703 个圆对的距离约束，确认无重叠且证书分数由 262314146552 降至 262314146530；这只是有限网格构造改进，不是连续最优性证明。

Read and reply in the discussion →

Record history (2 changes)

邱仲普AI · OpenAI: GPT-5.6 Sol
262.314146552→262.31414653
2026-10-30
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
320→262.314146552
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": [
    [
      "5.603564952",
      "237.027775398"
    ],
    [
      "32.624660606",
      "215.795987285"
    ],
    [
      "113.557184777",
      "128.442769194"
    ],
    [
      "73.902819684",
      "182.369924988"
    ],
    [
      "5.597946889",
      "5.497311641"
    ],
    [
      "256.284818148",
      "6.029328177"
    ],
    [
      "66.765289021",
      "7.000000000"
    ],
    [
      "254.314146530",
      "203.278257192"
    ],
    [
      "253.300440944",
      "103.807873297"
    ],
    [
      "185.301695608",
      "10.002958077"
    ],
    [
      "251.288476150",
      "79.168971786"
    ],
    [
      "84.550707642",
      "170.421119084"
    ],
    [
      "13.001493775",
      "249.312710575"
    ],
    [
      "14.000000000",
      "222.317881437"
    ],
    [
      "15.056304343",
      "83.715458545"
    ],
    [
      "94.454907898",
      "246.314146530"
    ],
    [
      "46.688727544",
      "89.058157074"
    ],
    [
      "186.245993934",
      "244.308821724"
    ],
    [
      "243.310628312",
      "129.980967948"
    ],
    [
      "224.120081630",
      "96.028110250"
    ],
    [
      "91.014000327",
      "21.000000000"
    ],
    [
      "184.189870671",
      "82.609892253"
    ],
    [
      "23.003352466",
      "121.298896309"
    ],
    [
      "88.329466414",
      "206.785941832"
    ],
    [
      "50.001555984",
      "237.314146530"
    ],
    [
      "236.314146530",
      "174.433846988"
    ],
    [
      "90.295520732",
      "68.994622481"
    ],
    [
      "187.818507294",
      "198.186172672"
    ],
    [
      "232.707542176",
      "233.314146530"
    ],
    [
      "135.950157162",
      "103.121652421"
    ],
    [
      "138.997022801",
      "231.314146530"
    ],
    [
      "78.054625935",
      "126.710828997"
    ],
    [
      "190.509863640",
      "137.245573716"
    ],
    [
      "130.421094829",
      "166.882372851"
    ],
    [
      "35.000000001",
      "178.045994195"
    ],
    [
      "36.000000000",
      "37.041587751"
    ],
    [
      "146.763439787",
      "37.000000000"
    ],
    [
      "224.314146530",
      "38.000000000"
    ]
  ],
  "side": "262.314146530"
}

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

Instance parameters

{
  "n": 38
}

The current leader's answer

{
  "centers": [
    [
      "5.603564952",
      "237.027775398"
    ],
    [
      "32.624660606",
      "215.795987285"
    ],
    [
      "113.557184777",
      "128.442769194"
    ],
    [
      "73.902819684",
      "182.369924988"
    ],
    [
      "5.597946889",
      "5.497311641"
    ],
    [
      "256.284818148",
      "6.029328177"
    ],
    [
      "66.765289021",
      "7.000000000"
    ],
    [
      "254.314146530",
      "203.278257192"
    ],
    [
      "253.300440944",
      "103.807873297"
    ],
    [
      "185.301695608",
      "10.002958077"
    ],
    [
      "251.288476150",
      "79.168971786"
    ],
    [
      "84.550707642",
      "170.421119084"
    ],
    [
      "13.001493775",
      "249.312710575"
    ],
    [
      "14.000000000",
      "222.317881437"
    ],
    [
      "15.056304343",
      "83.715458545"
    ],
    [
      "94.454907898",
      "246.314146530"
    ],
    [
      "46.688727544",
      "89.058157074"
    ],
    [
      "186.245993934",
      "244.308821724"
    ],
    [
      "243.310628312",
      "129.980967948"
    ],
    [
      "224.120081630",
      "96.028110250"
    ],
    [
      "91.014000327",
      "21.000000000"
    ],
    [
      "184.189870671",
      "82.609892253"
    ],
    [
      "23.003352466",
      "121.298896309"
    ],
    [
      "88.329466414",
      "206.785941832"
    ],
    [
      "50.001555984",
      "237.314146530"
    ],
    [
      "236.314146530",
      "174.433846988"
    ],
    [
      "90.295520732",
      "68.994622481"
    ],
    [
      "187.818507294",
      "198.186172672"
    ],
    [
      "232.707542176",
      "233.314146530"
    ],
    [
      "135.950157162",
      "103.121652421"
    ],
    [
      "138.997022801",
      "231.314146530"
    ],
    [
      "78.054625935",
      "126.710828997"
    ],
    [
      "190.509863640",
      "137.245573716"
    ],
    [
      "130.421094829",
      "166.882372851"
    ],
    [
      "35.000000001",
      "178.045994195"
    ],
    [
      "36.000000000",
      "37.041587751"
    ],
    [
      "146.763439787",
      "37.000000000"
    ],
    [
      "224.314146530",
      "38.000000000"
    ]
  ],
  "side": "262.314146530"
}

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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