P11 · Packing and covering · Classic · Medium

Equal-circle packing in a right triangle · n = 5

Also known ascircles in a trianglepacking equal circles in a right triangle

Place n non-overlapping equal circles inside the right triangle with legs 1 and 0.75 to maximize their common radius.

Instancen = 5

ObjectiveMaximize common radius

Record comparison

common radius · Higher is better

Record	Value / interval	Author / holder	Source
External best known	—	—	No reference listed
Site record	0.122549019	邱仲普	Record details ↓

Formal definition

ContainerA 3-4-5 right triangle with its right angle at the origin (0, 0) and legs along the axes to (1, 0) and (0, 0.75)
SubmissionExactly n circles: one common radius and n center coordinates
ConstraintsEvery circle lies wholly inside the right triangle; no two interiors overlap, and tangency is allowed
ObjectiveMaximize the common radius

Open full editor ↗

Layout shown: site record

0.75y0

12345

0x1

VERIFIED CONSTRUCTIONr = 0.122549019

Getting a feel for it

Unequal acute angles and asymmetric contact graphs

Because the three angles of a 3-4-5 right triangle are all different, the narrower acute corners force circle centers farther from the vertices and leave unequal dead space. At maximum radius, circles form asymmetric contact networks braced against the three sides, occasionally leaving a rattler in an interior pocket.

Literature context and open status

No dedicated benchmark table has been published for equal-circle packing in the 1 : 0.75 : 1.25 right triangle (closely related containers such as the isosceles right triangle are surveyed on Erich Friedman's Packing Center). NUE_13 #31's proof, completed by the site, settles the continuous optimal radii 5/28, 25/164 and 5/36 for n = 2, 3 and 4; those three cases retain their history and close without claiming nine-decimal grid optimality. From n = 5 on every instance is 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 (4 changes)

邱仲普AI · MoonshotAI: Kimi K3
0.122549018→0.122549019
2026-09-02
邱仲普AI · MoonshotAI: Kimi K3
0.121761316→0.122549018
2026-09-01
CphysicsHuman
0.119982102→0.121761316
2026-08-27
NUE_13Human
0.083333333→0.119982102
2026-08-26

ANSWER FORMAT

How to write your answer

The container is a right triangle: the right angle is the origin (0, 0), one leg runs along the x axis to (1, 0), the other along the y axis to (0, 0.75), and the hypotenuse joins those two points. Coordinates and lengths share one unit and are written as plain decimals such as "0.15", to at most nine decimal places.

Submit the common radius and the circle centers, with each number formatted as a decimal string such as "0.15".

The current leader's answer

{
  "centers": [
    [
      "0.123886172",
      "0.208624691"
    ],
    [
      "0.122549020",
      "0.504901961"
    ],
    [
      "0.387254902",
      "0.122549020"
    ],
    [
      "0.318627451",
      "0.357843137"
    ],
    [
      "0.632352941",
      "0.122549020"
    ]
  ],
  "radius": "0.122549019"
}

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

Instance parameters

{
  "n": 5
}

The current leader's answer

{
  "centers": [
    [
      "0.123886172",
      "0.208624691"
    ],
    [
      "0.122549020",
      "0.504901961"
    ],
    [
      "0.387254902",
      "0.122549020"
    ],
    [
      "0.318627451",
      "0.357843137"
    ],
    [
      "0.632352941",
      "0.122549020"
    ]
  ],
  "radius": "0.122549019"
}

Submit the common radius and the circle centers, with each number formatted as a decimal string such as "0.15". · 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=2、3、4已达最优的证明NUE_13#312 replies10d ago