P01 · Packing and covering · Classic

Equal-circle packing in a unit square · n = 28

Also known ascircle packing in a squarepacking equal circles in a unit squarecsq

Place n non-overlapping circles of equal radius inside the unit square to maximize their common radius.

Instancen = 28

ObjectiveMaximize common radius

Record comparison

common radius · Higher is better

Record	Value / interval	Author / holder	Source
Proven optimum	0.093672833	See source attribution	web.archive.org ↗
Site record	No player record yet	—	—

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

Formal definition

ContainerThe unit square [0, 1] × [0, 1], with lower-left corner (0, 0) and upper-right corner (1, 1)
SubmissionExactly n circles: one common radius and n center coordinates
ConstraintsEvery circle lies wholly inside the square; no two interiors overlap, and tangency is allowed
ObjectiveMaximize the common radius

Inspect in editor ↗

Layout shown: starter layout

1y0

12345678910111213141516171819202122232425262728

0x1

VERIFIED CONSTRUCTIONr = 0.093672833

Getting a feel for it

Contact graphs and irregular packings

At maximum radius, most circles are tangent to neighbors or the square boundary, forming a rigid contact network. Outside a few square numbers, regular grids waste space along the edges; better packings typically use tilted rows, staggered layers, and occasionally one or two rattlers that can move slightly inside a pocket.

Proven optima and open frontier

Mathematical optimality proofs are known for n = 1–30. For n = 31–300, the public best-known constructions from Packomania are imported; optimality remains open except at n = 36. Larger instances are grouped into a single table for algorithmic benchmarking.

Source

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.

ANSWER FORMAT

How to write your answer

The container is a square of side 1, from (0, 0) to (1, 1). Ordinary decimal certificates allow nine decimal places. For n ≤ 35, algebraic certificates allow fractions and one quadratic field; see the expression limits below.

Submit radius and centers as decimal strings. For n ≤ 35, algebraic certificates may add field: 2 or 3 and use fractions and the selected sqrt(2) or sqrt(3). Settled instances remain closed.

Starter answer

{
  "centers": [
    [
      "0.093672834",
      "0.093672834"
    ],
    [
      "0.389268333",
      "0.093672834"
    ],
    [
      "0.713760547",
      "0.093672834"
    ],
    [
      "0.901106215",
      "0.093672834"
    ],
    [
      "0.55151444",
      "0.187345668"
    ],
    [
      "0.241470583",
      "0.208799827"
    ],
    [
      "0.718981499",
      "0.280945739"
    ],
    [
      "0.906327166",
      "0.280945739"
    ],
    [
      "0.403716691",
      "0.302472661"
    ],
    [
      "0.093672834",
      "0.32392682"
    ],
    [
      "0.56820712",
      "0.392145945"
    ],
    [
      "0.255918941",
      "0.417599653"
    ],
    [
      "0.906327166",
      "0.468291406"
    ],
    [
      "0.721627368",
      "0.499666299"
    ],
    [
      "0.418165262",
      "0.51127212"
    ],
    [
      "0.093672834",
      "0.511272487"
    ],
    [
      "0.255918941",
      "0.604945321"
    ],
    [
      "0.574090162",
      "0.615126993"
    ],
    [
      "0.906327166",
      "0.655637074"
    ],
    [
      "0.093672834",
      "0.698618155"
    ],
    [
      "0.411844054",
      "0.708799827"
    ],
    [
      "0.730015275",
      "0.718981499"
    ],
    [
      "0.249597947",
      "0.802472661"
    ],
    [
      "0.567769167",
      "0.812654332"
    ],
    [
      "0.906327166",
      "0.842982742"
    ],
    [
      "0.093672834",
      "0.906327166"
    ],
    [
      "0.40552306",
      "0.906327166"
    ],
    [
      "0.730015275",
      "0.906327166"
    ]
  ],
  "radius": "0.093672833"
}

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

Instance parameters

{
  "n": 28
}

Starter answer

{
  "centers": [
    [
      "0.093672834",
      "0.093672834"
    ],
    [
      "0.389268333",
      "0.093672834"
    ],
    [
      "0.713760547",
      "0.093672834"
    ],
    [
      "0.901106215",
      "0.093672834"
    ],
    [
      "0.55151444",
      "0.187345668"
    ],
    [
      "0.241470583",
      "0.208799827"
    ],
    [
      "0.718981499",
      "0.280945739"
    ],
    [
      "0.906327166",
      "0.280945739"
    ],
    [
      "0.403716691",
      "0.302472661"
    ],
    [
      "0.093672834",
      "0.32392682"
    ],
    [
      "0.56820712",
      "0.392145945"
    ],
    [
      "0.255918941",
      "0.417599653"
    ],
    [
      "0.906327166",
      "0.468291406"
    ],
    [
      "0.721627368",
      "0.499666299"
    ],
    [
      "0.418165262",
      "0.51127212"
    ],
    [
      "0.093672834",
      "0.511272487"
    ],
    [
      "0.255918941",
      "0.604945321"
    ],
    [
      "0.574090162",
      "0.615126993"
    ],
    [
      "0.906327166",
      "0.655637074"
    ],
    [
      "0.093672834",
      "0.698618155"
    ],
    [
      "0.411844054",
      "0.708799827"
    ],
    [
      "0.730015275",
      "0.718981499"
    ],
    [
      "0.249597947",
      "0.802472661"
    ],
    [
      "0.567769167",
      "0.812654332"
    ],
    [
      "0.906327166",
      "0.842982742"
    ],
    [
      "0.093672834",
      "0.906327166"
    ],
    [
      "0.40552306",
      "0.906327166"
    ],
    [
      "0.730015275",
      "0.906327166"
    ]
  ],
  "radius": "0.093672833"
}

Submit radius and centers as decimal strings. For n ≤ 35, algebraic certificates may add field: 2 or 3 and use fractions and the selected sqrt(2) or sqrt(3). Settled instances remain closed. · Verifier v1.0.0

Algebraic certificates

Check exactly, then continue to the submission page for attribution and method. Only a formal submission saves the answer and enters the existing scoring system; feasibility does not prove optimality.

Use fractions, parentheses, arithmetic and sqrt(2) or sqrt(3). The field selects one root for the whole certificate; decimals are also accepted.

Complete layout (JSON)

{
  "field": 2,
  "radius": "1/12",
  "centers": [
    [
      "1/12",
      "1/12"
    ],
    [
      "3/12",
      "1/12"
    ],
    [
      "5/12",
      "1/12"
    ],
    [
      "7/12",
      "1/12"
    ],
    [
      "9/12",
      "1/12"
    ],
    [
      "11/12",
      "1/12"
    ],
    [
      "1/12",
      "3/12"
    ],
    [
      "3/12",
      "3/12"
    ],
    [
      "5/12",
      "3/12"
    ],
    [
      "7/12",
      "3/12"
    ],
    [
      "9/12",
      "3/12"
    ],
    [
      "11/12",
      "3/12"
    ],
    [
      "1/12",
      "5/12"
    ],
    [
      "3/12",
      "5/12"
    ],
    [
      "5/12",
      "5/12"
    ],
    [
      "7/12",
      "5/12"
    ],
    [
      "9/12",
      "5/12"
    ],
    [
      "11/12",
      "5/12"
    ],
    [
      "1/12",
      "7/12"
    ],
    [
      "3/12",
      "7/12"
    ],
    [
      "5/12",
      "7/12"
    ],
    [
      "7/12",
      "7/12"
    ],
    [
      "9/12",
      "7/12"
    ],
    [
      "11/12",
      "7/12"
    ],
    [
      "1/12",
      "9/12"
    ],
    [
      "3/12",
      "9/12"
    ],
    [
      "5/12",
      "9/12"
    ],
    [
      "7/12",
      "9/12"
    ]
  ]
}

Each expression: 256 characters, 64 tokens, 16 nesting levels; decimal literals have at most 18 decimal places and intermediate rational coefficients at most 256 bits.

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.

啊？？！NUE_13#312 replies28d ago 空悲切/.lird#282 replies45d ago