MinMax Arena
中/EN
P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 8 · n = 17

Pack n equal circles into the regular m-gon with centre (1,1), circumradius 1, and first vertex (2,1) so as to maximise their common radius.

Instancem = 8 · n = 17
ObjectiveMaximize circle radius

Record comparison

circle radius · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[0.198194125495, 0.198194125497]Eckard SpechtPackomania ↗
Site record0.198194125w33dRecord details ↓

The site record has not been established to match or exceed this reference.

Formal definition

  • ContainerThe editor and certificate normalise the container to the exact regular m-gon with centre (1,1), circumradius 1, and first vertex (2,1), while public scores follow the literature convention. Boundary contact is allowed.
  • 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.
  • ConstraintsExactly n congruent pieces lie wholly inside the container with pairwise disjoint interiors. Polygons may rotate independently by arbitrary angles.
  • PrecisionInput decimals are parsed as exact rationals and regular-polygon vertices are defined algebraically, with zero floating-point tolerance in feasibility checks.
  • ObjectiveMaximise the circle radius r (with container circumradius fixed at 1).
Open full editor ↗

Layout shown: site record

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. Pentagon and hexagon rows n = 41–100 and octagon rows n = 41–48 cite Packomania's value and author without importing its coordinates; they start from an ordinary site grid baseline.

Source
The record-holding arrangement for Equal circles in regular polygons m = 8 · n = 17, 0.198194125
LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holderw33d
Solution methodAI · Anthropic: Claude Fable 5.1
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)
  1. w33dAI · Anthropic: Claude Fable 5.1
    0.198194124→0.198194125
ANSWER FORMAT

How to write your answer

the regular m-gon with centre (1,1), circumradius 1, and first vertex (2,1)

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.

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.865956010",
      "y": "0.270046614"
    },
    {
      "turn": "0",
      "x": "0.490730215",
      "y": "0.427225774"
    },
    {
      "turn": "0",
      "x": "1.258953104",
      "y": "0.321785663"
    },
    {
      "turn": "0",
      "x": "0.306612856",
      "y": "0.781765124"
    },
    {
      "turn": "0",
      "x": "0.848109436",
      "y": "0.666032909"
    },
    {
      "turn": "0",
      "x": "1.592964008",
      "y": "0.535234396"
    },
    {
      "turn": "0",
      "x": "1.241106531",
      "y": "0.717771958"
    },
    {
      "turn": "0",
      "x": "0.644462638",
      "y": "1.006108822"
    },
    {
      "turn": "0",
      "x": "0.292893219",
      "y": "1.189200573"
    },
    {
      "turn": "0",
      "x": "1.037459732",
      "y": "1.057847871"
    },
    {
      "turn": "0",
      "x": "1.744655224",
      "y": "0.901449388"
    },
    {
      "turn": "0",
      "x": "0.796153854",
      "y": "1.372323814"
    },
    {
      "turn": "0",
      "x": "1.425860747",
      "y": "1.137020453"
    },
    {
      "turn": "0",
      "x": "0.444584435",
      "y": "1.555415565"
    },
    {
      "turn": "0",
      "x": "1.183131468",
      "y": "1.458184003"
    },
    {
      "turn": "0",
      "x": "0.915285556",
      "y": "1.750386353"
    },
    {
      "turn": "0",
      "x": "1.576921344",
      "y": "1.503496020"
    }
  ],
  "radius": "0.198194125"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 17,
  "sides": 8
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.865956010",
      "y": "0.270046614"
    },
    {
      "turn": "0",
      "x": "0.490730215",
      "y": "0.427225774"
    },
    {
      "turn": "0",
      "x": "1.258953104",
      "y": "0.321785663"
    },
    {
      "turn": "0",
      "x": "0.306612856",
      "y": "0.781765124"
    },
    {
      "turn": "0",
      "x": "0.848109436",
      "y": "0.666032909"
    },
    {
      "turn": "0",
      "x": "1.592964008",
      "y": "0.535234396"
    },
    {
      "turn": "0",
      "x": "1.241106531",
      "y": "0.717771958"
    },
    {
      "turn": "0",
      "x": "0.644462638",
      "y": "1.006108822"
    },
    {
      "turn": "0",
      "x": "0.292893219",
      "y": "1.189200573"
    },
    {
      "turn": "0",
      "x": "1.037459732",
      "y": "1.057847871"
    },
    {
      "turn": "0",
      "x": "1.744655224",
      "y": "0.901449388"
    },
    {
      "turn": "0",
      "x": "0.796153854",
      "y": "1.372323814"
    },
    {
      "turn": "0",
      "x": "1.425860747",
      "y": "1.137020453"
    },
    {
      "turn": "0",
      "x": "0.444584435",
      "y": "1.555415565"
    },
    {
      "turn": "0",
      "x": "1.183131468",
      "y": "1.458184003"
    },
    {
      "turn": "0",
      "x": "0.915285556",
      "y": "1.750386353"
    },
    {
      "turn": "0",
      "x": "1.576921344",
      "y": "1.503496020"
    }
  ],
  "radius": "0.198194125"
}

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.

本题m=6，n=3的证明NUE_13#310 replies