P84 · Packing and covering · Classic · Hard

Equilateral triangles in a disc · n = 17

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

Instancen = 17

ObjectiveMinimize container circle radius

Record comparison

container circle radius · Lower is better

Record	Value / interval	Author / holder	Source
External best known	[1.73, 1.731)	David W. Cantrell	Erich Friedman ↗
Site record	No 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

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": "0.119639667",
      "x": "1.222598852",
      "y": "1.758517176"
    },
    {
      "turn": "0.267731693",
      "x": "0.742727348",
      "y": "1.670722579"
    },
    {
      "turn": "-0.268166715",
      "x": "0.453694215",
      "y": "1.504005603"
    },
    {
      "turn": "-0.268166715",
      "x": "1.546714738",
      "y": "1.503561966"
    },
    {
      "turn": "-0.268166715",
      "x": "1.000085438",
      "y": "1.210500383"
    },
    {
      "turn": "0.267731693",
      "x": "0.591465461",
      "y": "1.170281105"
    },
    {
      "turn": "0.267731693",
      "x": "1.408672632",
      "y": "1.169949416"
    },
    {
      "turn": "-0.268166715",
      "x": "0.302432328",
      "y": "1.003564129"
    },
    {
      "turn": "-0.268166715",
      "x": "1.697570335",
      "y": "1.002997869"
    },
    {
      "turn": "0.267731693",
      "x": "0.999950008",
      "y": "0.876831860"
    },
    {
      "turn": "-0.268166715",
      "x": "0.710916875",
      "y": "0.710114883"
    },
    {
      "turn": "-0.268166715",
      "x": "1.288847712",
      "y": "0.709880312"
    },
    {
      "turn": "0.267731693",
      "x": "0.302296898",
      "y": "0.669895605"
    },
    {
      "turn": "0.267731693",
      "x": "1.697434906",
      "y": "0.669329345"
    },
    {
      "turn": "0.267731693",
      "x": "0.710781446",
      "y": "0.376446360"
    },
    {
      "turn": "0.267731693",
      "x": "1.288712282",
      "y": "0.376211788"
    },
    {
      "turn": "-0.268166715",
      "x": "0.999679149",
      "y": "0.209494812"
    }
  ],
  "radius": "0.333668550"
}

Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 17,
  "sides": 5
}

Starter answer

{
  "placements": [
    {
      "turn": "0.119639667",
      "x": "1.222598852",
      "y": "1.758517176"
    },
    {
      "turn": "0.267731693",
      "x": "0.742727348",
      "y": "1.670722579"
    },
    {
      "turn": "-0.268166715",
      "x": "0.453694215",
      "y": "1.504005603"
    },
    {
      "turn": "-0.268166715",
      "x": "1.546714738",
      "y": "1.503561966"
    },
    {
      "turn": "-0.268166715",
      "x": "1.000085438",
      "y": "1.210500383"
    },
    {
      "turn": "0.267731693",
      "x": "0.591465461",
      "y": "1.170281105"
    },
    {
      "turn": "0.267731693",
      "x": "1.408672632",
      "y": "1.169949416"
    },
    {
      "turn": "-0.268166715",
      "x": "0.302432328",
      "y": "1.003564129"
    },
    {
      "turn": "-0.268166715",
      "x": "1.697570335",
      "y": "1.002997869"
    },
    {
      "turn": "0.267731693",
      "x": "0.999950008",
      "y": "0.876831860"
    },
    {
      "turn": "-0.268166715",
      "x": "0.710916875",
      "y": "0.710114883"
    },
    {
      "turn": "-0.268166715",
      "x": "1.288847712",
      "y": "0.709880312"
    },
    {
      "turn": "0.267731693",
      "x": "0.302296898",
      "y": "0.669895605"
    },
    {
      "turn": "0.267731693",
      "x": "1.697434906",
      "y": "0.669329345"
    },
    {
      "turn": "0.267731693",
      "x": "0.710781446",
      "y": "0.376446360"
    },
    {
      "turn": "0.267731693",
      "x": "1.288712282",
      "y": "0.376211788"
    },
    {
      "turn": "-0.268166715",
      "x": "0.999679149",
      "y": "0.209494812"
    }
  ],
  "radius": "0.333668550"
}

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.