P12 · Packing and covering · Classic

Equal-circle packing in a 2:1 rectangle · n = 4

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 = 4

ObjectiveMaximize common radius

Record comparison

common radius · Higher is better

Record	Value / interval	Author / holder	Source
Proven optimum	(11 − 6√3)/2	See source attribution	Proof / source a zigzag of four, r = (11 − 6√3)/2
Site record	No player record yet	—	—

FinishedA construction reaching the theoretical ceiling is already published, leaving nothing further to improve

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

Inspect in editor ↗

Layout shown: starter layout

1y0

1234

0x2

VERIFIED CONSTRUCTIONr = 0.303847577

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.

LAYOUT & HISTORY

Starter layout

Inspect the layout, solver notes and record history.

The optimum is proven; no further improvement is possibleSubmit a proof / idea ↓Share a proof or bound argument in the discussion board. Accepted contributions earn proof points.

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".

Starter answer

{
  "centers": [
    [
      "0.303847577",
      "0.303847577"
    ],
    [
      "0.767949192",
      "0.696152423"
    ],
    [
      "1.232050807",
      "0.303847577"
    ],
    [
      "1.696152422",
      "0.696152423"
    ]
  ],
  "radius": "0.303847577"
}

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

Instance parameters

{
  "n": 4
}

Starter answer

{
  "centers": [
    [
      "0.303847577",
      "0.303847577"
    ],
    [
      "0.767949192",
      "0.696152423"
    ],
    [
      "1.232050807",
      "0.303847577"
    ],
    [
      "1.696152422",
      "0.696152423"
    ]
  ],
  "radius": "0.303847577"
}

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

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.

本题n=8时用抽屉原理的证明NUE_13#313 replies10d ago 好！！！NUE_13#310 replies17d ago