MinMax Arena
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P10 · Packing and covering · Formed here

Equal circles in a plus sign · n = 3

Place n non-overlapping circles of equal radius inside a symmetric cross (a 3 × 3 square with all four 1 × 1 corner squares removed) to maximize their common radius.

Instancen = 3
ObjectiveMaximize common radius

Record comparison

common radius · Higher is better
RecordValue / intervalAuthor / holderSource
Proven optimum1/2See source attributionMinMax Arena · Proof ↗
Site recordNo player record yet——

Formal definition

  • ContainerA symmetric cross-shaped region formed by removing the four 1 × 1 corner squares from [0, 3] × [0, 3], with lower-left corner (0, 0) and upper-right corner (3, 3)
  • SubmissionExactly n circles: one common radius and n center coordinates
  • ConstraintsEvery circle lies wholly inside the cross-shaped container; no two interiors overlap, and tangency is allowed
  • ObjectiveMaximize the common radius
Inspect in editor ↗

Layout shown: starter layout

3y0
0x3
VERIFIED CONSTRUCTIONr = 0.500000000

Getting a feel for it

Five unit squares and four re-entrant corners

The cross consists of a central 1 × 1 square surrounded by four 1 × 1 arms, meeting at four re-entrant right-angle corners. When n does not divide cleanly across the five unit squares, circles bridging the center and the arms must navigate the re-entrant corners, often breaking symmetry in the optimal contact graph.

Proven small cases and open frontier

For this cross-shaped packing problem, user #46 proved that the optimal common radius for n = 3, 4, and 5 is 1/2; those instances are closed as proven archives. No optimality proofs are known yet for n = 6…18, which remain open.

Source
The current arrangement for Equal circles in a plus sign 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.Read the published proof ↗
Record history (3 changes)
  1. MinMax ArenaHuman
    0.5→0.5
  2. NUE_13Human
    0.49→0.5
  3. NUE_13Human
    0.5→0.49
ANSWER FORMAT

How to write your answer

The container is a square of side 3 with all four of its 1 × 1 corners removed, leaving a plus sign: the origin (0, 0) is its lower-left corner and (3, 3) its upper-right. Coordinates and radii 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". All circles share a single radius.

Starter answer

{
  "centers": [
    [
      "1.5",
      "0.5"
    ],
    [
      "1.5",
      "1.5"
    ],
    [
      "2.5",
      "1.5"
    ]
  ],
  "radius": "0.5"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 3
}

Starter answer

{
  "centers": [
    [
      "1.5",
      "0.5"
    ],
    [
      "1.5",
      "1.5"
    ],
    [
      "2.5",
      "1.5"
    ]
  ],
  "radius": "0.5"
}

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

居然能发帖了(NUE_13#312 replies你们的思路是真清奇NUE_13#310 repliesn=2、3、4、5时的最优解的证明Anonymous#462 replies你好呀!LK今日里森堡#470 replies