MinMax Arena
中/EN
P83 · Packing and covering · Classic · Hard

Equilateral triangles in a square · n = 12

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

Instancen = 12
ObjectiveMinimize container square side

Record comparison

container square side · Lower is better
RecordValue / intervalAuthor / holderSource
External best known[2.558, 2.559)Dominik KampErich Friedman ↗
Site recordNo player record yet——

Formal definition

  • ContainerThe editor and certificate normalise the container to the unit square [0,1]², 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 square side = 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 square n = 12
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 unit square [0,1]²

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": "-0.566575612",
      "x": "0.225592998",
      "y": "0.802799051"
    },
    {
      "turn": "0.008118872",
      "x": "0.451185996",
      "y": "0.806462414"
    },
    {
      "turn": "-0.566575612",
      "x": "0.884030936",
      "y": "0.802799051"
    },
    {
      "turn": "0.008118872",
      "x": "0.115969064",
      "y": "0.605598103"
    },
    {
      "turn": "-0.566575612",
      "x": "0.548814004",
      "y": "0.601934740"
    },
    {
      "turn": "0.008118872",
      "x": "0.774407002",
      "y": "0.605598103"
    },
    {
      "turn": "-0.566575612",
      "x": "0.225592998",
      "y": "0.394401897"
    },
    {
      "turn": "0.008118872",
      "x": "0.451185996",
      "y": "0.398065260"
    },
    {
      "turn": "-0.566575612",
      "x": "0.884030936",
      "y": "0.394401897"
    },
    {
      "turn": "0.008118872",
      "x": "0.115969064",
      "y": "0.197200949"
    },
    {
      "turn": "-0.566575612",
      "x": "0.548814004",
      "y": "0.193537586"
    },
    {
      "turn": "0.008118872",
      "x": "0.774407002",
      "y": "0.197200949"
    }
  ],
  "radius": "0.225622739"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 12,
  "sides": 5
}

Starter answer

{
  "placements": [
    {
      "turn": "-0.566575612",
      "x": "0.225592998",
      "y": "0.802799051"
    },
    {
      "turn": "0.008118872",
      "x": "0.451185996",
      "y": "0.806462414"
    },
    {
      "turn": "-0.566575612",
      "x": "0.884030936",
      "y": "0.802799051"
    },
    {
      "turn": "0.008118872",
      "x": "0.115969064",
      "y": "0.605598103"
    },
    {
      "turn": "-0.566575612",
      "x": "0.548814004",
      "y": "0.601934740"
    },
    {
      "turn": "0.008118872",
      "x": "0.774407002",
      "y": "0.605598103"
    },
    {
      "turn": "-0.566575612",
      "x": "0.225592998",
      "y": "0.394401897"
    },
    {
      "turn": "0.008118872",
      "x": "0.451185996",
      "y": "0.398065260"
    },
    {
      "turn": "-0.566575612",
      "x": "0.884030936",
      "y": "0.394401897"
    },
    {
      "turn": "0.008118872",
      "x": "0.115969064",
      "y": "0.197200949"
    },
    {
      "turn": "-0.566575612",
      "x": "0.548814004",
      "y": "0.193537586"
    },
    {
      "turn": "0.008118872",
      "x": "0.774407002",
      "y": "0.197200949"
    }
  ],
  "radius": "0.225622739"
}

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.

证书归一化半径居然是中心到顶点的距离NUE_13#312 replies