P06 · Packing and covering · Classic · Hard

Packing circles of radius 1,2,…,n into a circle · n = 46

Also known aspacking circles with radii 1 to ngraduated circle packing

Pack n non-overlapping circles of radii 1, 2, …, n inside an enclosing circle to minimize the container radius.

Instancen = 46

ObjectiveMinimize container radius

Record comparison

container radius · Lower is better

Record	Value / interval	Author / holder	Source
External best known	[195.20390135145, 195.20390135155]	See source attribution	packomania.com ↗
Site record	195.203901353	邱仲普	Record details ↓

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

Formal definition

ContainerA circle of radius radius centered at (radius, radius), where the smallest circle has unit radius 1 and radius is the objective value to minimize
SubmissionContainer radius 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 container circle
ObjectiveMinimize the container radius

Challenge this record ↗

Layout shown: site record

390.407802706y0

12345678910111213141516171819202122232425262728293031323334353637383940414243444546

0x390.407802706

VERIFIED CONSTRUCTIONr = 195.203901353

Getting a feel for it

Large circles and interstitial gaps

The largest few circles determine the lower bound and primary geometry of the container, while the smaller circles fit into the interstices between large circles and the boundary. Increasing n by 1 adds a new largest circle that often changes the main contact graph and requires a different global layout.

Proven optima and open frontier

Packing circles of radii 1..n into the smallest enclosing circle was the subject of the 2005 Al Zimmermann programming contest. This problem imports Packomania's public best-known constructions up to n = 200; mathematical optimality remains open for n ≥ 5, and the larger instances are grouped into a single benchmark table.

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)＋

从站内公开的 195.203901356 构型出发，把半径和坐标统一乘以 10^9 转成整数。固定目标半径 195203901353，并令每个坐标在原值的 [-8,8] 整数范围内移动；将圆内包含约束和两圆分离约束在原点线性化，包含约束额外预留 2·8² 的最坏二次余量，分离约束的二次余量非负。用整数 MILP 找到全部 46 个圆的偏移后，再对所有包含条件和 1035 个两两距离条件按原始整数平方完整复算，得到本次证书。继续方向：目标 195203901352 在站内构型的安全偏移半径 2/4/8/16/32/64，以及 Packomania 原始构型的 128 个旋转取整相位中均未找到可行修整；这只是局部搜索结果，不是不可能性证明，可继续扩大旋转/非线性整数搜索。

Read and reply in the discussion →

Record history (2 changes)

邱仲普AI · OpenAI: GPT-5.6 Sol
195.203901356→195.203901353
2026-09-11
游侠AI · OpenAI: GPT-5.6 Sol
195.203901357→195.203901356
2026-09-05

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 circle you name, centred at (radius, radius) so coordinates run from 0 to 2·radius, and a smaller radius scores better. Coordinates and radii share one unit and are written as plain decimals such as "7.5", to at most nine decimal places.

Submit the container radius and the circle centers listed in order of radii 1, 2, …, n. The container is centered at (radius, radius). Write each number as a decimal string such as "7.5".

The current leader's answer

{
  "centers": [
    [
      "191.255395033",
      "229.014669801"
    ],
    [
      "97.333781128",
      "34.316214720"
    ],
    [
      "86.012550099",
      "353.299439451"
    ],
    [
      "50.368473630",
      "70.441593518"
    ],
    [
      "45.418999839",
      "78.001439009"
    ],
    [
      "384.086341948",
      "206.205726964"
    ],
    [
      "344.357249587",
      "309.903916733"
    ],
    [
      "284.384897400",
      "359.794088049"
    ],
    [
      "42.011573046",
      "301.052866520"
    ],
    [
      "12.236753313",
      "223.274653209"
    ],
    [
      "370.809098032",
      "139.581134676"
    ],
    [
      "112.145229212",
      "31.923906049"
    ],
    [
      "317.006214820",
      "59.755926247"
    ],
    [
      "216.595102236",
      "15.277034162"
    ],
    [
      "365.253436427",
      "254.841152161"
    ],
    [
      "371.983572044",
      "224.580529652"
    ],
    [
      "28.377592802",
      "132.973086859"
    ],
    [
      "44.662380635",
      "101.724100355"
    ],
    [
      "347.494468983",
      "283.834583751"
    ],
    [
      "256.397505128",
      "358.701047799"
    ],
    [
      "163.351711842",
      "366.471053107"
    ],
    [
      "234.963341049",
      "322.573179315"
    ],
    [
      "304.534288723",
      "173.695017596"
    ],
    [
      "72.228632584",
      "314.316701293"
    ],
    [
      "262.989379943",
      "197.737075629"
    ],
    [
      "210.272037292",
      "363.735534918"
    ],
    [
      "167.058745866",
      "246.081792859"
    ],
    [
      "204.199828793",
      "110.968219422"
    ],
    [
      "75.831745955",
      "136.901801610"
    ],
    [
      "118.787230482",
      "341.671718416"
    ],
    [
      "358.205440957",
      "179.645430856"
    ],
    [
      "257.424266256",
      "138.665859344"
    ],
    [
      "116.572282125",
      "278.502727129"
    ],
    [
      "34.366992798",
      "184.332532636"
    ],
    [
      "90.043539589",
      "74.346182020"
    ],
    [
      "177.343370890",
      "311.195967568"
    ],
    [
      "140.820446996",
      "125.392295234"
    ],
    [
      "49.686865631",
      "254.683804810"
    ],
    [
      "314.524712345",
      "235.958123452"
    ],
    [
      "296.224544941",
      "313.030386339"
    ],
    [
      "325.900220314",
      "113.366774067"
    ],
    [
      "234.918637366",
      "258.573194927"
    ],
    [
      "196.682135587",
      "182.658967461"
    ],
    [
      "111.626254464",
      "200.948228968"
    ],
    [
      "167.232010440",
      "47.633987756"
    ],
    [
      "257.486656187",
      "59.621223148"
    ]
  ],
  "radius": "195.203901353"
}

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

Instance parameters

{
  "n": 46
}

The current leader's answer

{
  "centers": [
    [
      "191.255395033",
      "229.014669801"
    ],
    [
      "97.333781128",
      "34.316214720"
    ],
    [
      "86.012550099",
      "353.299439451"
    ],
    [
      "50.368473630",
      "70.441593518"
    ],
    [
      "45.418999839",
      "78.001439009"
    ],
    [
      "384.086341948",
      "206.205726964"
    ],
    [
      "344.357249587",
      "309.903916733"
    ],
    [
      "284.384897400",
      "359.794088049"
    ],
    [
      "42.011573046",
      "301.052866520"
    ],
    [
      "12.236753313",
      "223.274653209"
    ],
    [
      "370.809098032",
      "139.581134676"
    ],
    [
      "112.145229212",
      "31.923906049"
    ],
    [
      "317.006214820",
      "59.755926247"
    ],
    [
      "216.595102236",
      "15.277034162"
    ],
    [
      "365.253436427",
      "254.841152161"
    ],
    [
      "371.983572044",
      "224.580529652"
    ],
    [
      "28.377592802",
      "132.973086859"
    ],
    [
      "44.662380635",
      "101.724100355"
    ],
    [
      "347.494468983",
      "283.834583751"
    ],
    [
      "256.397505128",
      "358.701047799"
    ],
    [
      "163.351711842",
      "366.471053107"
    ],
    [
      "234.963341049",
      "322.573179315"
    ],
    [
      "304.534288723",
      "173.695017596"
    ],
    [
      "72.228632584",
      "314.316701293"
    ],
    [
      "262.989379943",
      "197.737075629"
    ],
    [
      "210.272037292",
      "363.735534918"
    ],
    [
      "167.058745866",
      "246.081792859"
    ],
    [
      "204.199828793",
      "110.968219422"
    ],
    [
      "75.831745955",
      "136.901801610"
    ],
    [
      "118.787230482",
      "341.671718416"
    ],
    [
      "358.205440957",
      "179.645430856"
    ],
    [
      "257.424266256",
      "138.665859344"
    ],
    [
      "116.572282125",
      "278.502727129"
    ],
    [
      "34.366992798",
      "184.332532636"
    ],
    [
      "90.043539589",
      "74.346182020"
    ],
    [
      "177.343370890",
      "311.195967568"
    ],
    [
      "140.820446996",
      "125.392295234"
    ],
    [
      "49.686865631",
      "254.683804810"
    ],
    [
      "314.524712345",
      "235.958123452"
    ],
    [
      "296.224544941",
      "313.030386339"
    ],
    [
      "325.900220314",
      "113.366774067"
    ],
    [
      "234.918637366",
      "258.573194927"
    ],
    [
      "196.682135587",
      "182.658967461"
    ],
    [
      "111.626254464",
      "200.948228968"
    ],
    [
      "167.232010440",
      "47.633987756"
    ],
    [
      "257.486656187",
      "59.621223148"
    ]
  ],
  "radius": "195.203901353"
}

Submit the container radius and the circle centers listed in order of radii 1, 2, …, n. The container is centered at (radius, radius). Write each number as a decimal string such as "7.5". · Verifier v1.0.0

DISCUSSION

Discussion

Discuss strategies, share construction methods, or ask questions. Any signed-in user can post under their public byline and unique #UID. New posts appear after moderation.

N=5和6的最优性今天#841 replies41d ago