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

Equal circles in regular polygons · m = 6 · n = 33

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 = 6 · n = 33
ObjectiveMaximize circle radius

Record comparison

circle radius · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[0.142870299181, 0.142870299183]Eckard SpechtPackomania ↗
Site record0.142870299condorRecord 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 = 6 · n = 33, 0.142870299
LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holdercondor
Solution methodAI · OpenAI: GPT-6.1 Sol
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. condorAI · OpenAI: GPT-6.1 Sol
    0.142870298→0.142870299
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": "1.033199214",
      "y": "1.228237870"
    },
    {
      "turn": "0",
      "x": "1.131773196",
      "y": "0.771762130"
    },
    {
      "turn": "0",
      "x": "0.848155325",
      "y": "0.806526950"
    },
    {
      "turn": "0",
      "x": "1.243474912",
      "y": "1.034764822"
    },
    {
      "turn": "0",
      "x": "0.785740597",
      "y": "1.085367570"
    },
    {
      "turn": "0",
      "x": "0.890328915",
      "y": "1.475696487"
    },
    {
      "turn": "0",
      "x": "1.176069513",
      "y": "1.475696487"
    },
    {
      "turn": "0",
      "x": "0.988902897",
      "y": "0.524303513"
    },
    {
      "turn": "0",
      "x": "1.417513794",
      "y": "0.771762130"
    },
    {
      "turn": "0",
      "x": "0.705285026",
      "y": "0.559068333"
    },
    {
      "turn": "0",
      "x": "1.274643495",
      "y": "0.524303513"
    },
    {
      "turn": "0",
      "x": "0.562414727",
      "y": "0.806526950"
    },
    {
      "turn": "0",
      "x": "1.386345211",
      "y": "1.282223439"
    },
    {
      "turn": "0",
      "x": "0.499999999",
      "y": "1.085367570"
    },
    {
      "turn": "0",
      "x": "1.529215510",
      "y": "1.034764822"
    },
    {
      "turn": "0",
      "x": "0.642870298",
      "y": "1.332826187"
    },
    {
      "turn": "0",
      "x": "1.033199214",
      "y": "1.723155104"
    },
    {
      "turn": "0",
      "x": "0.499999999",
      "y": "1.580284804"
    },
    {
      "turn": "0",
      "x": "1.318939812",
      "y": "1.723155104"
    },
    {
      "turn": "0",
      "x": "1.131773196",
      "y": "0.276844896"
    },
    {
      "turn": "0",
      "x": "1.703254392",
      "y": "0.771762130"
    },
    {
      "turn": "0",
      "x": "0.846032598",
      "y": "0.276844896"
    },
    {
      "turn": "0",
      "x": "1.560384093",
      "y": "0.524303513"
    },
    {
      "turn": "0",
      "x": "0.562414727",
      "y": "0.311609716"
    },
    {
      "turn": "0",
      "x": "0.419544428",
      "y": "0.559068333"
    },
    {
      "turn": "0",
      "x": "1.417513794",
      "y": "0.276844896"
    },
    {
      "turn": "0",
      "x": "1.529215510",
      "y": "1.529682056"
    },
    {
      "turn": "0",
      "x": "0.276674129",
      "y": "0.806526950"
    },
    {
      "turn": "0",
      "x": "1.672085809",
      "y": "1.282223439"
    },
    {
      "turn": "0",
      "x": "0.214259401",
      "y": "1.085367570"
    },
    {
      "turn": "0",
      "x": "0.357129700",
      "y": "1.332826187"
    },
    {
      "turn": "0",
      "x": "1.814956108",
      "y": "1.034764822"
    },
    {
      "turn": "0",
      "x": "0.747458616",
      "y": "1.723155104"
    }
  ],
  "radius": "0.142870299"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 33,
  "sides": 6
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "1.033199214",
      "y": "1.228237870"
    },
    {
      "turn": "0",
      "x": "1.131773196",
      "y": "0.771762130"
    },
    {
      "turn": "0",
      "x": "0.848155325",
      "y": "0.806526950"
    },
    {
      "turn": "0",
      "x": "1.243474912",
      "y": "1.034764822"
    },
    {
      "turn": "0",
      "x": "0.785740597",
      "y": "1.085367570"
    },
    {
      "turn": "0",
      "x": "0.890328915",
      "y": "1.475696487"
    },
    {
      "turn": "0",
      "x": "1.176069513",
      "y": "1.475696487"
    },
    {
      "turn": "0",
      "x": "0.988902897",
      "y": "0.524303513"
    },
    {
      "turn": "0",
      "x": "1.417513794",
      "y": "0.771762130"
    },
    {
      "turn": "0",
      "x": "0.705285026",
      "y": "0.559068333"
    },
    {
      "turn": "0",
      "x": "1.274643495",
      "y": "0.524303513"
    },
    {
      "turn": "0",
      "x": "0.562414727",
      "y": "0.806526950"
    },
    {
      "turn": "0",
      "x": "1.386345211",
      "y": "1.282223439"
    },
    {
      "turn": "0",
      "x": "0.499999999",
      "y": "1.085367570"
    },
    {
      "turn": "0",
      "x": "1.529215510",
      "y": "1.034764822"
    },
    {
      "turn": "0",
      "x": "0.642870298",
      "y": "1.332826187"
    },
    {
      "turn": "0",
      "x": "1.033199214",
      "y": "1.723155104"
    },
    {
      "turn": "0",
      "x": "0.499999999",
      "y": "1.580284804"
    },
    {
      "turn": "0",
      "x": "1.318939812",
      "y": "1.723155104"
    },
    {
      "turn": "0",
      "x": "1.131773196",
      "y": "0.276844896"
    },
    {
      "turn": "0",
      "x": "1.703254392",
      "y": "0.771762130"
    },
    {
      "turn": "0",
      "x": "0.846032598",
      "y": "0.276844896"
    },
    {
      "turn": "0",
      "x": "1.560384093",
      "y": "0.524303513"
    },
    {
      "turn": "0",
      "x": "0.562414727",
      "y": "0.311609716"
    },
    {
      "turn": "0",
      "x": "0.419544428",
      "y": "0.559068333"
    },
    {
      "turn": "0",
      "x": "1.417513794",
      "y": "0.276844896"
    },
    {
      "turn": "0",
      "x": "1.529215510",
      "y": "1.529682056"
    },
    {
      "turn": "0",
      "x": "0.276674129",
      "y": "0.806526950"
    },
    {
      "turn": "0",
      "x": "1.672085809",
      "y": "1.282223439"
    },
    {
      "turn": "0",
      "x": "0.214259401",
      "y": "1.085367570"
    },
    {
      "turn": "0",
      "x": "0.357129700",
      "y": "1.332826187"
    },
    {
      "turn": "0",
      "x": "1.814956108",
      "y": "1.034764822"
    },
    {
      "turn": "0",
      "x": "0.747458616",
      "y": "1.723155104"
    }
  ],
  "radius": "0.142870299"
}

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