MinMax Arena
中/EN
P84 · Packing and covering · Classic · Hard

Equilateral triangles in a disc · n = 7

Pack n unit-side equilateral triangles into a disc. Contact is allowed; interiors cannot overlap. Minimise the container circle radius.

Instancen = 7
ObjectiveMinimize container circle radius

Record comparison

container circle radius · Lower is better
RecordValue / intervalAuthor / holderSource
External best known[1.152, 1.153)Erich FriedmanErich Friedman ↗
Site recordNo player record yet——

Formal definition

  • ContainerThe editor and certificate normalise the container to the closed unit disc centred at (1,1), while public scores are reported in the literature's unit-piece scale. 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.
  • ObjectiveContainer circle radius = 1/(2 sin(π/3) r), where smaller is better. Pages, leaderboards, and literature records share this unit with automatic conversion.
Open full 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 Equilateral triangles in a disc n = 7
LAYOUT & HISTORY

Starter layout

Inspect the layout, solver notes and record history.

Challenge this record ↗Submit 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": "0.151017308",
      "x": "1.144380713",
      "y": "1.634239855"
    },
    {
      "turn": "-0.481223484",
      "x": "0.594151216",
      "y": "1.508323468"
    },
    {
      "turn": "-0.308146982",
      "x": "1.585888588",
      "y": "1.282560695"
    },
    {
      "turn": "0.000285071",
      "x": "0.349534131",
      "y": "0.999629142"
    },
    {
      "turn": "0.308771371",
      "x": "1.586210406",
      "y": "0.718107568"
    },
    {
      "turn": "0.481925850",
      "x": "0.594731113",
      "y": "0.491214080"
    },
    {
      "turn": "-0.150434214",
      "x": "1.145103832",
      "y": "0.365925192"
    }
  ],
  "radius": "0.501005786"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 7,
  "sides": 5
}

Starter answer

{
  "placements": [
    {
      "turn": "0.151017308",
      "x": "1.144380713",
      "y": "1.634239855"
    },
    {
      "turn": "-0.481223484",
      "x": "0.594151216",
      "y": "1.508323468"
    },
    {
      "turn": "-0.308146982",
      "x": "1.585888588",
      "y": "1.282560695"
    },
    {
      "turn": "0.000285071",
      "x": "0.349534131",
      "y": "0.999629142"
    },
    {
      "turn": "0.308771371",
      "x": "1.586210406",
      "y": "0.718107568"
    },
    {
      "turn": "0.481925850",
      "x": "0.594731113",
      "y": "0.491214080"
    },
    {
      "turn": "-0.150434214",
      "x": "1.145103832",
      "y": "0.365925192"
    }
  ],
  "radius": "0.501005786"
}

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.