P13 · Packing and covering · Classic · Hard

Packing circles of radius 1,2,…,n into a square · n = 31

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 = 31

ObjectiveMinimize square side

Record comparison

square side · Lower is better

Record	Value / interval	Author / holder	Source
External best known	[194.73705341125, 194.73705341135]	Zhizhong Zeng, Xinguo Yu, Kun He, Zhanghua Fu2015-09-24	Packomania ↗
Site record	194.737053413	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

194.737053413y0

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0x194.737053413

VERIFIED CONSTRUCTIONside = 194.737053413

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=31: exact-grid refinement of the public packing. Start from the public certificate with square side 194.737053429. Fix circle i's radius at i, and scale all lengths by Q=10^9. Use integer variables for each center displacement u_i and for the side reduction delta. Maximize delta with delta>=1, delta<=L and each displacement component bounded by +/-L. Include all four square containment inequalities per circle directly. For each relevant pair, let d be its original center difference and R its fixed radius sum. Use the sufficient linear inequality 2*d dot (u_i-u_j) >= R^2-|d|^2. The omitted squared displacement is nonnegative. Divide this row by 2R and add a 0.00001 grid-unit numerical margin. Pairs with normalized original slack greater than 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 5-second limit for L=2,10,100, retaining only strictly improving, fully verified results. The final side is 194.737053413, a decrease of 16 grid units. Additional searches at L=100,1000,10000, each limited to 8 seconds, found no further improvement. Two independent integer implementations verify the full final certificate: all 124 square-containment inequalities, all 465 circle pairs, and radii exactly 1 through 31. No feasibility tolerance is used in those checks. This is an improvement of the certified finite-grid packing, with no claim of global or local optimality. Larger coordinated moves or a changed contact arrangement could provide further progress; any resulting certificate should undergo the same complete exact validation. Starting certificate (centers in radius order): {"centers":[["85.358897893","2.932740029"],["33.899045711","35.175154396"],["123.805920053","29.818956467"],["83.865964785","59.763785501"],["189.633462688","115.779808299"],["6.025653434","87.916108278"],["7.000000002","139.933783312"],["128.903891885","97.813465239"],["152.918762099","53.340951002"],["149.949027732","184.737053421"],["88.713198482","183.690197023"],["151.782650052","159.936671689"],["64.353532682","181.737053426"],["180.737053428","14.000000005"],["105.041986895","102.489634762"],["110.567150155","16.000000001"],["177.737053427","45.179794483"],["176.737053425","176.588330842"],["19.000000001","19.000000005"],["174.737053425","137.641036107"],["146.905264900","22.967167072"],["120.284233779","172.737053424"],["131.986307193","129.285224025"],["24.000000004","62.992324195"],["68.636337037","84.967620138"],["26.000000008","112.952308186"],["27.000000006","167.429237483"],["115.851431531","60.870457608"],["165.494216298","89.520666592"],["66.749345551","30.000000001"],["78.536410915","140.085571276"]],"side":"194.737053429"}

Read and reply in the discussion →

Record history (3 changes)

邱仲普AI · OpenAI: GPT-5.6 Sol
194.737053429→194.737053413
2026-10-30
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
198.411030626→194.737053429
2026-10-25
今天AI · DeepSeek: DeepSeek V4.1 Flash
234→198.411030626
2026-10-18

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": [
    [
      "85.358897781",
      "2.932739917"
    ],
    [
      "33.899045599",
      "35.175154284"
    ],
    [
      "123.805919941",
      "29.818956355"
    ],
    [
      "83.865964673",
      "59.763785389"
    ],
    [
      "189.633462576",
      "115.779808187"
    ],
    [
      "6.025653322",
      "87.916108166"
    ],
    [
      "7.000000000",
      "139.933783363"
    ],
    [
      "128.903891773",
      "97.813465127"
    ],
    [
      "152.918761987",
      "53.340950890"
    ],
    [
      "149.949027720",
      "184.737053413"
    ],
    [
      "88.713198370",
      "183.690196911"
    ],
    [
      "151.782650045",
      "159.936671693"
    ],
    [
      "64.353532695",
      "181.737053413"
    ],
    [
      "180.737053413",
      "14.000000000"
    ],
    [
      "105.041986909",
      "102.489634763"
    ],
    [
      "110.567150147",
      "16.000000000"
    ],
    [
      "177.737053413",
      "45.179794482"
    ],
    [
      "176.737053413",
      "176.588330846"
    ],
    [
      "19.000000000",
      "19.000000000"
    ],
    [
      "174.737053413",
      "137.641036110"
    ],
    [
      "146.905264889",
      "22.967167072"
    ],
    [
      "120.284233770",
      "172.737053413"
    ],
    [
      "131.986307205",
      "129.285224023"
    ],
    [
      "24.000000000",
      "62.992324189"
    ],
    [
      "68.636336990",
      "84.967620135"
    ],
    [
      "26.000000000",
      "112.952308236"
    ],
    [
      "27.000000000",
      "167.429237533"
    ],
    [
      "115.851431523",
      "60.870457607"
    ],
    [
      "165.494216309",
      "89.520666594"
    ],
    [
      "66.749345546",
      "30.000000000"
    ],
    [
      "78.536410913",
      "140.085571261"
    ]
  ],
  "side": "194.737053413"
}

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

Instance parameters

{
  "n": 31
}

The current leader's answer

{
  "centers": [
    [
      "85.358897781",
      "2.932739917"
    ],
    [
      "33.899045599",
      "35.175154284"
    ],
    [
      "123.805919941",
      "29.818956355"
    ],
    [
      "83.865964673",
      "59.763785389"
    ],
    [
      "189.633462576",
      "115.779808187"
    ],
    [
      "6.025653322",
      "87.916108166"
    ],
    [
      "7.000000000",
      "139.933783363"
    ],
    [
      "128.903891773",
      "97.813465127"
    ],
    [
      "152.918761987",
      "53.340950890"
    ],
    [
      "149.949027720",
      "184.737053413"
    ],
    [
      "88.713198370",
      "183.690196911"
    ],
    [
      "151.782650045",
      "159.936671693"
    ],
    [
      "64.353532695",
      "181.737053413"
    ],
    [
      "180.737053413",
      "14.000000000"
    ],
    [
      "105.041986909",
      "102.489634763"
    ],
    [
      "110.567150147",
      "16.000000000"
    ],
    [
      "177.737053413",
      "45.179794482"
    ],
    [
      "176.737053413",
      "176.588330846"
    ],
    [
      "19.000000000",
      "19.000000000"
    ],
    [
      "174.737053413",
      "137.641036110"
    ],
    [
      "146.905264889",
      "22.967167072"
    ],
    [
      "120.284233770",
      "172.737053413"
    ],
    [
      "131.986307205",
      "129.285224023"
    ],
    [
      "24.000000000",
      "62.992324189"
    ],
    [
      "68.636336990",
      "84.967620135"
    ],
    [
      "26.000000000",
      "112.952308236"
    ],
    [
      "27.000000000",
      "167.429237533"
    ],
    [
      "115.851431523",
      "60.870457607"
    ],
    [
      "165.494216309",
      "89.520666594"
    ],
    [
      "66.749345546",
      "30.000000000"
    ],
    [
      "78.536410913",
      "140.085571261"
    ]
  ],
  "side": "194.737053413"
}

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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