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

Equal discs covering a disc · n = 30

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

Instancen = 30
ObjectiveMaximize covered disc radius

Record comparison

covered disc radius · Higher is better
RecordValue / intervalAuthor / holderSource
External best known——No reference listed
Site record4.575512899NUE_13Record details ↓

Formal definition

  • ContainerThe editor and certificate normalize to the closed unit disc centred at (1,1). This is a coordinate convention only; scores use the literature's unit-piece scale. Boundary contact is allowed.
  • SubmissionSubmit {radius, placements:[{x,y,turn},…]}, with decimal strings of at most nine places. x,y are centres; circles require turn=0, polygons use turn=tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4.
  • ConstraintsThe n distinct centres lie in the target disc. Covering discs may overlap and protrude, but must cover every target point.
  • PrecisionInput decimals are exact rationals; regular-polygon vertices are algebraic, not rounded templates. Feasibility has no floating-point tolerance. A finite-decimal certificate is not a continuous optimality proof.
  • ObjectiveCovered disc radius R = 1/r, larger is better.
Open the full editor ↗

Layout shown: site record

VERIFIED CONSTRUCTIONVERIFIED CONSTRUCTION

Getting a feel for it

Source and certificate

External targets are converted with their printed precision; decimal coordinate certificates are separate. Reconstruction loss never lowers the literature target, and a source construction is not automatically an optimality proof.

Source
The record-holding arrangement for Equal discs covering a disc n = 30, 4.575512899
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 idea in the discussion. Accepted contributions can 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)
  1. NUE_13Human
    4.575499877→4.575512899
ANSWER FORMAT

How to write your answer

the closed unit disc centred at (1,1)

Submit {radius, placements:[{x,y,turn},…]}, with decimal strings of at most nine places. x,y are centres; circles require turn=0, polygons use turn=tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4.

The current leader's answer

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "1.150357553",
      "y": "0.864784670"
    },
    {
      "turn": "0.000000000",
      "x": "0.850978261",
      "y": "1.140098146"
    },
    {
      "turn": "0.000000000",
      "x": "1.199546078",
      "y": "1.225015639"
    },
    {
      "turn": "0.000000000",
      "x": "0.803566591",
      "y": "0.780735364"
    },
    {
      "turn": "0.000000000",
      "x": "1.500809419",
      "y": "0.954796721"
    },
    {
      "turn": "0.000000000",
      "x": "0.895341425",
      "y": "1.497916508"
    },
    {
      "turn": "0.000000000",
      "x": "1.106855375",
      "y": "0.505929946"
    },
    {
      "turn": "0.000000000",
      "x": "0.498936351",
      "y": "1.045563062"
    },
    {
      "turn": "0.000000000",
      "x": "1.457827898",
      "y": "0.601484817"
    },
    {
      "turn": "0.000000000",
      "x": "0.542017597",
      "y": "1.399415655"
    },
    {
      "turn": "0.000000000",
      "x": "1.544982223",
      "y": "1.308255849"
    },
    {
      "turn": "0.000000000",
      "x": "1.238920716",
      "y": "1.582754276"
    },
    {
      "turn": "0.000000000",
      "x": "0.762575010",
      "y": "0.421344698"
    },
    {
      "turn": "0.000000000",
      "x": "0.455963521",
      "y": "0.689961723"
    },
    {
      "turn": "0.000000000",
      "x": "1.820195166",
      "y": "0.723282696"
    },
    {
      "turn": "0.000000000",
      "x": "0.631963953",
      "y": "1.774737183"
    },
    {
      "turn": "0.000000000",
      "x": "1.847871623",
      "y": "1.026132929"
    },
    {
      "turn": "0.000000000",
      "x": "1.365454702",
      "y": "0.224399177"
    },
    {
      "turn": "0.000000000",
      "x": "0.926935335",
      "y": "1.854049386"
    },
    {
      "turn": "0.000000000",
      "x": "0.179393707",
      "y": "1.274004869"
    },
    {
      "turn": "0.000000000",
      "x": "1.072808665",
      "y": "0.149285519"
    },
    {
      "turn": "0.000000000",
      "x": "0.153997517",
      "y": "0.972894055"
    },
    {
      "turn": "0.000000000",
      "x": "1.724276178",
      "y": "0.350426052"
    },
    {
      "turn": "0.000000000",
      "x": "1.591685077",
      "y": "1.659448961"
    },
    {
      "turn": "0.000000000",
      "x": "0.273046231",
      "y": "1.647569155"
    },
    {
      "turn": "0.000000000",
      "x": "1.876278563",
      "y": "1.392778790"
    },
    {
      "turn": "0.000000000",
      "x": "1.295897593",
      "y": "1.913600809"
    },
    {
      "turn": "0.000000000",
      "x": "0.407130868",
      "y": "0.337538986"
    },
    {
      "turn": "0.000000000",
      "x": "0.706822595",
      "y": "0.085867837"
    },
    {
      "turn": "0.000000000",
      "x": "0.122987433",
      "y": "0.607301131"
    }
  ],
  "radius": "0.218554733"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 30,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "1.150357553",
      "y": "0.864784670"
    },
    {
      "turn": "0.000000000",
      "x": "0.850978261",
      "y": "1.140098146"
    },
    {
      "turn": "0.000000000",
      "x": "1.199546078",
      "y": "1.225015639"
    },
    {
      "turn": "0.000000000",
      "x": "0.803566591",
      "y": "0.780735364"
    },
    {
      "turn": "0.000000000",
      "x": "1.500809419",
      "y": "0.954796721"
    },
    {
      "turn": "0.000000000",
      "x": "0.895341425",
      "y": "1.497916508"
    },
    {
      "turn": "0.000000000",
      "x": "1.106855375",
      "y": "0.505929946"
    },
    {
      "turn": "0.000000000",
      "x": "0.498936351",
      "y": "1.045563062"
    },
    {
      "turn": "0.000000000",
      "x": "1.457827898",
      "y": "0.601484817"
    },
    {
      "turn": "0.000000000",
      "x": "0.542017597",
      "y": "1.399415655"
    },
    {
      "turn": "0.000000000",
      "x": "1.544982223",
      "y": "1.308255849"
    },
    {
      "turn": "0.000000000",
      "x": "1.238920716",
      "y": "1.582754276"
    },
    {
      "turn": "0.000000000",
      "x": "0.762575010",
      "y": "0.421344698"
    },
    {
      "turn": "0.000000000",
      "x": "0.455963521",
      "y": "0.689961723"
    },
    {
      "turn": "0.000000000",
      "x": "1.820195166",
      "y": "0.723282696"
    },
    {
      "turn": "0.000000000",
      "x": "0.631963953",
      "y": "1.774737183"
    },
    {
      "turn": "0.000000000",
      "x": "1.847871623",
      "y": "1.026132929"
    },
    {
      "turn": "0.000000000",
      "x": "1.365454702",
      "y": "0.224399177"
    },
    {
      "turn": "0.000000000",
      "x": "0.926935335",
      "y": "1.854049386"
    },
    {
      "turn": "0.000000000",
      "x": "0.179393707",
      "y": "1.274004869"
    },
    {
      "turn": "0.000000000",
      "x": "1.072808665",
      "y": "0.149285519"
    },
    {
      "turn": "0.000000000",
      "x": "0.153997517",
      "y": "0.972894055"
    },
    {
      "turn": "0.000000000",
      "x": "1.724276178",
      "y": "0.350426052"
    },
    {
      "turn": "0.000000000",
      "x": "1.591685077",
      "y": "1.659448961"
    },
    {
      "turn": "0.000000000",
      "x": "0.273046231",
      "y": "1.647569155"
    },
    {
      "turn": "0.000000000",
      "x": "1.876278563",
      "y": "1.392778790"
    },
    {
      "turn": "0.000000000",
      "x": "1.295897593",
      "y": "1.913600809"
    },
    {
      "turn": "0.000000000",
      "x": "0.407130868",
      "y": "0.337538986"
    },
    {
      "turn": "0.000000000",
      "x": "0.706822595",
      "y": "0.085867837"
    },
    {
      "turn": "0.000000000",
      "x": "0.122987433",
      "y": "0.607301131"
    }
  ],
  "radius": "0.218554733"
}

Submit {radius, placements:[{x,y,turn},…]}, with decimal strings of at most nine places. x,y are centres; circles require turn=0, polygons use turn=tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4. · Verifier v1.0.0

DISCUSSION

Discussion

Talk strategy, share methods, ask why you are stuck. Every signed-in user can post. Posts carry your public byline, the same name your records use; the #number after it is the account's signup ordinal, so a name cannot be worn by someone else. New posts appear after an automated review.

No threads yet. Be the first to talk about this problem?