P10 · Packing and covering · Formed here · Medium

Equal circles in a plus sign · n = 15

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 = 15

ObjectiveMaximize common radius

Record comparison

common radius · Higher is better

Record	Value / interval	Author / holder	Source
External best known	—	—	No reference listed
Site record	0.27053504	邱仲普	Record details ↓

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

Open full editor ↗

Layout shown: site record

3y0

123456789101112131415

0x3

VERIFIED CONSTRUCTIONr = 0.270535040

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

LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holder邱仲普

Solution methodAI · MoonshotAI: Kimi K3

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 (9 changes)

邱仲普AI · MoonshotAI: Kimi K3
0.27→0.27053504
2026-09-01
HwaterBHuman
0.264961164→0.27
2026-08-30
NUE_13Human
0.263757264→0.264961164
2026-08-27
lirdHuman
0.263446615→0.263757264
2026-08-27
NUE_13Human
0.260695984→0.263446615
2026-08-27
EraserHuman
0.258419806→0.260695984
2026-08-27
lirdHuman
0.258146419→0.258419806
2026-08-27
lirdHuman
0.257911361→0.258146419
2026-08-27
NUE_13Human
0.2→0.257911361
2026-08-27

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.

The current leader's answer

{
  "centers": [
    [
      "0.843737148",
      "1.729464959"
    ],
    [
      "1.747966377",
      "1.901672684"
    ],
    [
      "1.625311852",
      "1.374688148"
    ],
    [
      "2.156262852",
      "1.270535041"
    ],
    [
      "2.729464959",
      "1.270535041"
    ],
    [
      "1.270535041",
      "2.729464959"
    ],
    [
      "1.270535041",
      "0.557136095"
    ],
    [
      "0.557136095",
      "1.270535041"
    ],
    [
      "1.729464959",
      "2.442863905"
    ],
    [
      "1.270535041",
      "2.156262852"
    ],
    [
      "2.442863905",
      "1.729464959"
    ],
    [
      "0.270535041",
      "1.729464959"
    ],
    [
      "1.729464959",
      "0.270535041"
    ],
    [
      "1.098327316",
      "1.252033623"
    ],
    [
      "1.729464959",
      "0.843737148"
    ]
  ],
  "radius": "0.270535040"
}

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

Instance parameters

{
  "n": 15
}

The current leader's answer

{
  "centers": [
    [
      "0.843737148",
      "1.729464959"
    ],
    [
      "1.747966377",
      "1.901672684"
    ],
    [
      "1.625311852",
      "1.374688148"
    ],
    [
      "2.156262852",
      "1.270535041"
    ],
    [
      "2.729464959",
      "1.270535041"
    ],
    [
      "1.270535041",
      "2.729464959"
    ],
    [
      "1.270535041",
      "0.557136095"
    ],
    [
      "0.557136095",
      "1.270535041"
    ],
    [
      "1.729464959",
      "2.442863905"
    ],
    [
      "1.270535041",
      "2.156262852"
    ],
    [
      "2.442863905",
      "1.729464959"
    ],
    [
      "0.270535041",
      "1.729464959"
    ],
    [
      "1.729464959",
      "0.270535041"
    ],
    [
      "1.098327316",
      "1.252033623"
    ],
    [
      "1.729464959",
      "0.843737148"
    ]
  ],
  "radius": "0.270535040"
}

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 replies65d ago 你们的思路是真清奇NUE_13#310 replies65d ago n=2、3、4、5时的最优解的证明Anonymous#462 replies65d ago 你好呀!LK今日里森堡#470 replies65d ago