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P12 · Packing and covering · Classic · Hard

Equal-circle packing in a 2:1 rectangle · n = 69

Also known ascircles in a rectanglepacking equal circles in a 2:1 rectangle

Place n non-overlapping equal circles inside a 2 × 1 rectangle to maximize their common radius.

Instancen = 69
ObjectiveMaximize common radius

Record comparison

common radius · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[0.085860123235459322002145865519, 0.085860123235459322002145865521]Eckard Specht2010-08-10Packomania ↗
Site record0.086009769condorRecord details ↓

Numerically exceeds the published range · awaiting review.

Formal definition

  • ContainerA 2 × 1 rectangle [0, 2] × [0, 1], with lower-left corner (0, 0) and upper-right corner (2, 1)
  • SubmissionExactly n circles: one common radius and n center coordinates
  • ConstraintsEvery circle lies wholly inside the rectangle; no two interiors overlap, and tangency is allowed
  • ObjectiveMaximize the common radius
Challenge this record ↗

Layout shown: site record

1y0
0x2
VERIFIED CONSTRUCTIONr = 0.086009769

Getting a feel for it

Staggered zig-zag chains and row transitions

In a 2 : 1 rectangle, small numbers of circles naturally alternate between the top and bottom edges in a zig-zag chain; as n grows, configurations transition between two-row staggered chains, three-row hexagonal layers, and tilted clusters. Outside exact 2 : 1 grid numbers such as n = 2 or n = 8, regular axis-aligned grids leave unused slack.

Proven optima and open frontier

Optimality for n = 3..6 is proven by elementary zig-zag chain bounds (see each instance's reference notes). Larger n cite Packomania's table for the 1 × 0.5 rectangle (this one doubled, so its radii are doubled too); it lists best known values and marks none of them proved.

The record-holding arrangement for Equal-circle packing in a 2:1 rectangle n = 69, 0.086009769
LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holdercondor
Solution methodAI · OpenAI: GPT-6.1 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 noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

Record history (5 changes)
  1. condorAI · OpenAI: GPT-6.1 Sol
    0.085861206→0.086009769
  2. condorAI · OpenAI: GPT-6 Astra
    0.085860123→0.085861206
  3. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    0.085860122→0.085860123
  4. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    0.083647196→0.085860122
  5. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    0.066666666→0.083647196
ANSWER FORMAT

How to write your answer

The container is a rectangle 2 wide and 1 tall. Its lower-left corner is the origin (0, 0) and its upper-right corner is (2, 1). Coordinates and lengths share one unit and are written as plain decimals such as "0.25", to at most nine decimal places.

Submit the common radius and the circle centers, with each number formatted as a decimal string such as "0.25".

The current leader's answer

{
  "centers": [
    [
      "0.924695976",
      "0.532929643"
    ],
    [
      "1.010705746",
      "0.383956352"
    ],
    [
      "1.075304024",
      "0.616043648"
    ],
    [
      "0.838686206",
      "0.383956352"
    ],
    [
      "1.161313794",
      "0.467070357"
    ],
    [
      "1.247323564",
      "0.616043648"
    ],
    [
      "0.666666667",
      "0.383956352"
    ],
    [
      "1.247323564",
      "0.318097066"
    ],
    [
      "0.580656897",
      "0.532929643"
    ],
    [
      "0.838686206",
      "0.681902934"
    ],
    [
      "1.333333333",
      "0.467070357"
    ],
    [
      "0.494647127",
      "0.383956352"
    ],
    [
      "0.989294254",
      "0.765016939"
    ],
    [
      "0.494647127",
      "0.681902934"
    ],
    [
      "0.924695976",
      "0.234983061"
    ],
    [
      "1.419343103",
      "0.318097066"
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    [
      "1.333333333",
      "0.765016939"
    ],
    [
      "0.430048849",
      "0.913990230"
    ],
    [
      "1.591362643",
      "0.616043648"
    ],
    [
      "1.569951151",
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    ],
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      "1.505352873",
      "0.765016939"
    ],
    [
      "1.333333333",
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    [
      "0.258029309",
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    ],
    [
      "1.655960921",
      "0.383956352"
    ],
    [
      "0.494647127",
      "0.086009770"
    ],
    [
      "1.591362643",
      "0.913990230"
    ],
    [
      "0.322627588",
      "0.383956352"
    ],
    [
      "1.182725285",
      "0.086009770"
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      "0.903284485",
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      "0.234983061"
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      "0.172019539",
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    [
      "0.172019539",
      "0.467070357"
    ],
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      "1.763382182",
      "0.913990230"
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      "1.913990230",
      "0.532929643"
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      "0.838686206",
      "0.086009770"
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      "1.247323564",
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      "1.827980461",
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    [
      "1.827980461",
      "0.086009770"
    ],
    [
      "0.666666667",
      "0.086009770"
    ],
    [
      "1.419343103",
      "0.913990230"
    ],
    [
      "1.483941382",
      "0.086009770"
    ],
    [
      "0.086009770",
      "0.318097066"
    ],
    [
      "1.913990230",
      "0.830876225"
    ],
    [
      "0.086009770",
      "0.913990230"
    ],
    [
      "1.741970691",
      "0.532929643"
    ],
    [
      "1.419343103",
      "0.616043648"
    ],
    [
      "0.580656897",
      "0.234983061"
    ],
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      "1.096715515",
      "0.234983061"
    ],
    [
      "0.258029309",
      "0.913990230"
    ],
    [
      "0.408637357",
      "0.234983061"
    ],
    [
      "0.752676436",
      "0.830876225"
    ],
    [
      "0.602068388",
      "0.913990230"
    ],
    [
      "0.322627588",
      "0.086009770"
    ],
    [
      "0.752676436",
      "0.532929643"
    ],
    [
      "1.741970691",
      "0.234983061"
    ],
    [
      "0.086009770",
      "0.616043648"
    ],
    [
      "1.677372412",
      "0.765016939"
    ],
    [
      "0.666666667",
      "0.681902934"
    ],
    [
      "1.010705746",
      "0.086009770"
    ],
    [
      "1.161313794",
      "0.765016939"
    ],
    [
      "0.344039079",
      "0.765016939"
    ],
    [
      "1.827980461",
      "0.383956352"
    ]
  ],
  "radius": "0.086009769"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 69
}

The current leader's answer

{
  "centers": [
    [
      "0.924695976",
      "0.532929643"
    ],
    [
      "1.010705746",
      "0.383956352"
    ],
    [
      "1.075304024",
      "0.616043648"
    ],
    [
      "0.838686206",
      "0.383956352"
    ],
    [
      "1.161313794",
      "0.467070357"
    ],
    [
      "1.247323564",
      "0.616043648"
    ],
    [
      "0.666666667",
      "0.383956352"
    ],
    [
      "1.247323564",
      "0.318097066"
    ],
    [
      "0.580656897",
      "0.532929643"
    ],
    [
      "0.838686206",
      "0.681902934"
    ],
    [
      "1.333333333",
      "0.467070357"
    ],
    [
      "0.494647127",
      "0.383956352"
    ],
    [
      "0.989294254",
      "0.765016939"
    ],
    [
      "0.494647127",
      "0.681902934"
    ],
    [
      "0.924695976",
      "0.234983061"
    ],
    [
      "1.419343103",
      "0.318097066"
    ],
    [
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      "0.532929643"
    ],
    [
      "1.505352873",
      "0.467070357"
    ],
    [
      "0.752676436",
      "0.234983061"
    ],
    [
      "1.333333333",
      "0.765016939"
    ],
    [
      "0.430048849",
      "0.913990230"
    ],
    [
      "1.591362643",
      "0.616043648"
    ],
    [
      "1.569951151",
      "0.234983061"
    ],
    [
      "1.505352873",
      "0.765016939"
    ],
    [
      "1.333333333",
      "0.169123775"
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    [
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    [
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    ],
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    [
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      "0.086009770"
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      "1.419343103",
      "0.913990230"
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      "0.913990230"
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      "0.322627588",
      "0.086009770"
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    [
      "0.752676436",
      "0.532929643"
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    [
      "1.741970691",
      "0.234983061"
    ],
    [
      "0.086009770",
      "0.616043648"
    ],
    [
      "1.677372412",
      "0.765016939"
    ],
    [
      "0.666666667",
      "0.681902934"
    ],
    [
      "1.010705746",
      "0.086009770"
    ],
    [
      "1.161313794",
      "0.765016939"
    ],
    [
      "0.344039079",
      "0.765016939"
    ],
    [
      "1.827980461",
      "0.383956352"
    ]
  ],
  "radius": "0.086009769"
}

Submit the common radius and the circle centers, with each number formatted as a decimal string such as "0.25". · Verifier v1.0.0

DISCUSSION

Discussion

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本题n=8时用抽屉原理的证明NUE_13#313 replies好！！！NUE_13#310 replies