P08 · Packing and covering · Classic · Hard

Equal circles in an L · n = 12

Place n non-overlapping circles of equal radius inside the L-shaped region to maximize their common radius.

Instancen = 12

ObjectiveMaximize common radius

Record comparison

common radius · Higher is better

Record	Value / interval	Author / holder	Source
External best known	1/4	Trivial grid	Erich Friedman ↗
Site record	0.25	NUE_13	Record details ↓

Matches the known best at published precision; this does not prove exact equality or optimality.

Formal definition

ContainerAn L-shaped region formed by removing the top-right 1 × 1 square (1, 2] × (1, 2] from [0, 2] × [0, 2], with the origin (0, 0) at the lower-left corner
SubmissionExactly n circles: one common radius and n center coordinates
ConstraintsEvery circle lies wholly inside the L-shaped region (staying clear of the re-entrant corner at (1, 1) and the notch); no two interiors overlap, and tangency is allowed
ObjectiveMaximize the common radius

Open full editor ↗

Layout shown: site record

2y0

123456789101112

0x2

VERIFIED CONSTRUCTIONr = 0.250000000

Getting a feel for it

Re-entrant corner and contact graphs

The L-shaped container consists of three 1 × 1 squares meeting at the re-entrant corner (1, 1), which pinches circles transitioning between the two arms. At maximum radius, circles lock into a rigid contact network against the outer walls, the re-entrant corner, and one another, making the allocation of circles across the three unit squares decisive.

Literature benchmarks and proven cases

Erich Friedman's Circles in L's survey records constructions and contributors dating back to 2005. Whereas the source fixes unit-radius circles and minimizes the outer side s of the L, this problem fixes the outer side at 2, relating the common radius by r = 2/s. Optimality for n = 4 has been proven and closed on this site; for n = 5–18, external literature targets and current site records are listed alongside each other.

Source

LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holderNUE_13

Solution methodHuman

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 (1 changes)

NUE_13Human
0.083333333→0.25
2026-08-26

ANSWER FORMAT

How to write your answer

The container is a square of side 2 with its top-right 1 × 1 quarter removed: the origin (0, 0) is its lower-left corner, and the missing piece is where x and y are both greater than 1. 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": [
    [
      "0.250000000",
      "1.750000000"
    ],
    [
      "0.750000000",
      "1.750000000"
    ],
    [
      "0.250000000",
      "1.250000000"
    ],
    [
      "0.750000000",
      "1.250000000"
    ],
    [
      "0.250000000",
      "0.750000000"
    ],
    [
      "0.750000000",
      "0.750000000"
    ],
    [
      "0.250000000",
      "0.250000000"
    ],
    [
      "0.750000000",
      "0.250000000"
    ],
    [
      "1.250000000",
      "0.750000000"
    ],
    [
      "1.750000000",
      "0.750000000"
    ],
    [
      "1.250000000",
      "0.250000000"
    ],
    [
      "1.750000000",
      "0.250000000"
    ]
  ],
  "radius": "0.250000000"
}

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

Instance parameters

{
  "n": 12
}

The current leader's answer

{
  "centers": [
    [
      "0.250000000",
      "1.750000000"
    ],
    [
      "0.750000000",
      "1.750000000"
    ],
    [
      "0.250000000",
      "1.250000000"
    ],
    [
      "0.750000000",
      "1.250000000"
    ],
    [
      "0.250000000",
      "0.750000000"
    ],
    [
      "0.750000000",
      "0.750000000"
    ],
    [
      "0.250000000",
      "0.250000000"
    ],
    [
      "0.750000000",
      "0.250000000"
    ],
    [
      "1.250000000",
      "0.750000000"
    ],
    [
      "1.750000000",
      "0.750000000"
    ],
    [
      "1.250000000",
      "0.250000000"
    ],
    [
      "1.750000000",
      "0.250000000"
    ]
  ],
  "radius": "0.250000000"
}

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.

本题n=8时的证明（抽屉原理）NUE_13#314 replies10d ago 问题等价，有外部记录NUE_13#311 replies45d ago 问题等价，有外部记录NUE_13#311 replies45d ago