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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 5 · n = 24

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 = 5 · n = 24
ObjectiveMaximize circle radius

Record comparison

circle radius · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[0.156027298449, 0.156027298451]Eckard SpechtPackomania ↗
Site record0.156027298w33dRecord 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 = 5 · n = 24, 0.156027298
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
    0.156027297→0.156027298
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": "0.347010304",
      "y": "1.189641929"
    },
    {
      "turn": "0",
      "x": "0.347010304",
      "y": "0.770252756"
    },
    {
      "turn": "0",
      "x": "0.410717068",
      "y": "1.495124367"
    },
    {
      "turn": "0",
      "x": "0.472838293",
      "y": "0.484691224"
    },
    {
      "turn": "0",
      "x": "0.578108231",
      "y": "0.979947343"
    },
    {
      "turn": "0",
      "x": "0.643419238",
      "y": "1.287211472"
    },
    {
      "turn": "0",
      "x": "0.703936220",
      "y": "0.694385810"
    },
    {
      "turn": "0",
      "x": "0.709471113",
      "y": "1.592195440"
    },
    {
      "turn": "0",
      "x": "0.769920432",
      "y": "0.388382585"
    },
    {
      "turn": "0",
      "x": "0.874517164",
      "y": "1.077516885"
    },
    {
      "turn": "0",
      "x": "0.940569040",
      "y": "1.382500854"
    },
    {
      "turn": "0",
      "x": "1.000345153",
      "y": "0.791955353"
    },
    {
      "turn": "0",
      "x": "1.006252671",
      "y": "1.688625614"
    },
    {
      "turn": "0",
      "x": "1.066687873",
      "y": "0.486054187"
    },
    {
      "turn": "0",
      "x": "1.186246724",
      "y": "1.063277755"
    },
    {
      "turn": "0",
      "x": "1.249375819",
      "y": "0.232586963"
    },
    {
      "turn": "0",
      "x": "1.252298600",
      "y": "1.368261724"
    },
    {
      "turn": "0",
      "x": "1.312074713",
      "y": "0.777716223"
    },
    {
      "turn": "0",
      "x": "1.317932470",
      "y": "1.673335922"
    },
    {
      "turn": "0",
      "x": "1.433196988",
      "y": "0.485859555"
    },
    {
      "turn": "0",
      "x": "1.496671510",
      "y": "1.031426162"
    },
    {
      "turn": "0",
      "x": "1.562723386",
      "y": "1.336410131"
    },
    {
      "turn": "0",
      "x": "1.622499499",
      "y": "0.745864630"
    },
    {
      "turn": "0",
      "x": "1.807139653",
      "y": "1.000000000"
    }
  ],
  "radius": "0.156027298"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 24,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.347010304",
      "y": "1.189641929"
    },
    {
      "turn": "0",
      "x": "0.347010304",
      "y": "0.770252756"
    },
    {
      "turn": "0",
      "x": "0.410717068",
      "y": "1.495124367"
    },
    {
      "turn": "0",
      "x": "0.472838293",
      "y": "0.484691224"
    },
    {
      "turn": "0",
      "x": "0.578108231",
      "y": "0.979947343"
    },
    {
      "turn": "0",
      "x": "0.643419238",
      "y": "1.287211472"
    },
    {
      "turn": "0",
      "x": "0.703936220",
      "y": "0.694385810"
    },
    {
      "turn": "0",
      "x": "0.709471113",
      "y": "1.592195440"
    },
    {
      "turn": "0",
      "x": "0.769920432",
      "y": "0.388382585"
    },
    {
      "turn": "0",
      "x": "0.874517164",
      "y": "1.077516885"
    },
    {
      "turn": "0",
      "x": "0.940569040",
      "y": "1.382500854"
    },
    {
      "turn": "0",
      "x": "1.000345153",
      "y": "0.791955353"
    },
    {
      "turn": "0",
      "x": "1.006252671",
      "y": "1.688625614"
    },
    {
      "turn": "0",
      "x": "1.066687873",
      "y": "0.486054187"
    },
    {
      "turn": "0",
      "x": "1.186246724",
      "y": "1.063277755"
    },
    {
      "turn": "0",
      "x": "1.249375819",
      "y": "0.232586963"
    },
    {
      "turn": "0",
      "x": "1.252298600",
      "y": "1.368261724"
    },
    {
      "turn": "0",
      "x": "1.312074713",
      "y": "0.777716223"
    },
    {
      "turn": "0",
      "x": "1.317932470",
      "y": "1.673335922"
    },
    {
      "turn": "0",
      "x": "1.433196988",
      "y": "0.485859555"
    },
    {
      "turn": "0",
      "x": "1.496671510",
      "y": "1.031426162"
    },
    {
      "turn": "0",
      "x": "1.562723386",
      "y": "1.336410131"
    },
    {
      "turn": "0",
      "x": "1.622499499",
      "y": "0.745864630"
    },
    {
      "turn": "0",
      "x": "1.807139653",
      "y": "1.000000000"
    }
  ],
  "radius": "0.156027298"
}

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

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本题m=6，n=3的证明NUE_13#310 replies