MinMax Arena
中/EN
P87 · Packing and covering · Classic

Equal discs covering a disc · n = 18

Cover the largest possible disc with n unit-radius discs. Maximise the covered disc radius.

Instancen = 18
ObjectiveMaximize covered disc radius

Record comparison

covered disc radius · Higher is better
RecordValue / intervalAuthor / holderSource
Proven optimum[3.446282750304803656149542229383, 3.446282750304803656149542229384]VonEquinoxDiskCoveringSolve ↗
Site recordNo player record yet——

The continuous optimum is proved, so competition submissions are closed. Historical records and contribution points are retained. Because finite-decimal coordinates cannot attain this value exactly, the current certificate is not claimed optimal on the nine-decimal grid.

Formal definition

  • ContainerThe editor and certificate normalise the target disc to the closed unit disc centred at (1,1), while public scores are reported in the literature's unit-covering-disc scale.
  • SubmissionSubmit {radius, placements:[{x,y,turn},…]}, with each value written as a decimal string of at most nine places. Here x, y are centre coordinates; circles require turn = 0 and polygons use turn = tan(θ/2). Coordinates and turn must lie in [−4, 4], with 0 < radius ≤ 4.
  • ConstraintsThe n distinct centres lie inside the target disc. Covering discs may overlap and protrude, and their union must cover every point of the target disc.
  • PrecisionInput decimals are parsed as exact rationals, with zero floating-point tolerance in feasibility checks.
  • ObjectiveCovered disc radius R = 1/r, where larger is better.
Inspect in editor ↗

Layout shown: starter layout

VERIFIED CONSTRUCTIONVERIFIED CONSTRUCTION

Getting a feel for it

Literature targets and coordinate certificates

Literature targets are converted at their published precision, while nine-decimal coordinate certificates are stored and displayed alongside the source values.

Source
The current arrangement for Equal discs covering a disc n = 18
LAYOUT & HISTORY

Starter layout

Inspect the layout, solver notes and record history.

Proven optimumSubmit 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 closed unit disc centred at (1,1)

Submit {radius, placements:[{x,y,turn},…]}, with each value written as a decimal string of at most nine places. Here x, y are centre coordinates; circles require turn = 0 and polygons use turn = tan(θ/2). Coordinates and turn must lie in [−4, 4], with 0 < radius ≤ 4.

Starter answer

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "1.161848766",
      "y": "1.372346197"
    },
    {
      "turn": "0.000000000",
      "x": "1.174729055",
      "y": "0.633521508"
    },
    {
      "turn": "0.000000000",
      "x": "1.345116238",
      "y": "1.006016528"
    },
    {
      "turn": "0.000000000",
      "x": "0.694603983",
      "y": "1.337087782"
    },
    {
      "turn": "0.000000000",
      "x": "0.706539167",
      "y": "0.652472022"
    },
    {
      "turn": "0.000000000",
      "x": "0.797535415",
      "y": "0.996470359"
    },
    {
      "turn": "0.000000000",
      "x": "0.259546127",
      "y": "0.987091301"
    },
    {
      "turn": "0.000000000",
      "x": "1.694887681",
      "y": "1.414759953"
    },
    {
      "turn": "0.000000000",
      "x": "1.708922519",
      "y": "0.609713350"
    },
    {
      "turn": "0.000000000",
      "x": "1.000091094",
      "y": "0.189470838"
    },
    {
      "turn": "0.000000000",
      "x": "0.176887719",
      "y": "1.369956558"
    },
    {
      "turn": "0.000000000",
      "x": "0.190283181",
      "y": "0.601577471"
    },
    {
      "turn": "0.000000000",
      "x": "0.971838899",
      "y": "1.810039824"
    },
    {
      "turn": "0.000000000",
      "x": "0.509437275",
      "y": "1.766026483"
    },
    {
      "turn": "0.000000000",
      "x": "0.536436332",
      "y": "0.217339755"
    },
    {
      "turn": "0.000000000",
      "x": "1.864767794",
      "y": "1.015075957"
    },
    {
      "turn": "0.000000000",
      "x": "1.434189068",
      "y": "1.795750217"
    },
    {
      "turn": "0.000000000",
      "x": "1.461662162",
      "y": "0.219867658"
    }
  ],
  "radius": "0.290167718"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 18,
  "sides": 5
}

Starter answer

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "1.161848766",
      "y": "1.372346197"
    },
    {
      "turn": "0.000000000",
      "x": "1.174729055",
      "y": "0.633521508"
    },
    {
      "turn": "0.000000000",
      "x": "1.345116238",
      "y": "1.006016528"
    },
    {
      "turn": "0.000000000",
      "x": "0.694603983",
      "y": "1.337087782"
    },
    {
      "turn": "0.000000000",
      "x": "0.706539167",
      "y": "0.652472022"
    },
    {
      "turn": "0.000000000",
      "x": "0.797535415",
      "y": "0.996470359"
    },
    {
      "turn": "0.000000000",
      "x": "0.259546127",
      "y": "0.987091301"
    },
    {
      "turn": "0.000000000",
      "x": "1.694887681",
      "y": "1.414759953"
    },
    {
      "turn": "0.000000000",
      "x": "1.708922519",
      "y": "0.609713350"
    },
    {
      "turn": "0.000000000",
      "x": "1.000091094",
      "y": "0.189470838"
    },
    {
      "turn": "0.000000000",
      "x": "0.176887719",
      "y": "1.369956558"
    },
    {
      "turn": "0.000000000",
      "x": "0.190283181",
      "y": "0.601577471"
    },
    {
      "turn": "0.000000000",
      "x": "0.971838899",
      "y": "1.810039824"
    },
    {
      "turn": "0.000000000",
      "x": "0.509437275",
      "y": "1.766026483"
    },
    {
      "turn": "0.000000000",
      "x": "0.536436332",
      "y": "0.217339755"
    },
    {
      "turn": "0.000000000",
      "x": "1.864767794",
      "y": "1.015075957"
    },
    {
      "turn": "0.000000000",
      "x": "1.434189068",
      "y": "1.795750217"
    },
    {
      "turn": "0.000000000",
      "x": "1.461662162",
      "y": "0.219867658"
    }
  ],
  "radius": "0.290167718"
}

Submit {radius, placements:[{x,y,turn},…]}, with each value written as a decimal string of at most nine places. Here x, y are centre coordinates; circles require turn = 0 and polygons use turn = tan(θ/2). Coordinates and turn must lie in [−4, 4], with 0 < radius ≤ 4. · 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.

No threads yet. Start the first discussion.