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P06 · Packing and covering · Classic

Packing circles of radius 1,2,…,n into a circle · n = 3

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 = 3
ObjectiveMinimize container radius

Record comparison

container radius · Lower is better
RecordValue / intervalAuthor / holderSource
Proven optimum5See source attributionpackomania.com ↗
Site recordNo player record yet——

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

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
Inspect in editor ↗

Layout shown: starter layout

10y0
0x10
VERIFIED CONSTRUCTIONr = 5

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
The current arrangement for Packing circles of radius 1,2,…,n into a circle n = 3
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.
Record history (3 changes)
  1. Packomania · Eckard SpechtHuman
    5→5
  2. SneakyZeroHuman
    5.243323455→5
  3. AnonymousHuman
    5→5.243323455
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".

Starter answer

{
  "centers": [
    [
      "6",
      "8"
    ],
    [
      "8",
      "5"
    ],
    [
      "3",
      "5"
    ]
  ],
  "radius": "5"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 3
}

Starter answer

{
  "centers": [
    [
      "6",
      "8"
    ],
    [
      "8",
      "5"
    ],
    [
      "3",
      "5"
    ]
  ],
  "radius": "5"
}

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

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N=5和6的最优性今天#841 replies