Fully cover a rectangle with minimum amount of fixed radius circles

Question

I\'ve had this problem for a few years. It was on an informatics contest in my town a while back. I failed to solve it, and my teacher failed to solve it. I haven\'t met anyone

Answer 1

The hexagon is better than the diamond. Consider the percent area of the unit circle covered by each:

#!/usr/bin/env ruby

include Math

def diamond
  # The distance from the center to a corner is the radius.
  # On a unit circle, that is 1.
  radius = 1

  # The edge of the nested diamond is the hypotenuse of a
  # right triangle whose legs are both radii.
  edge = sqrt(radius ** 2 + radius ** 2)

  # The area of the diamond is the square of the edge
  edge ** 2
end

def hexagon
  # The hexagon is composed of 6 equilateral triangles.
  # Since the inner edges go from the center to a hexagon
  # corner, their length is the radius (1).
  radius = 1

  # The base and height of an equilateral triangle whose
  # edge is 'radius'.
  base = radius
  height = sin(PI / 3) * radius

  # The area of said triangle
  triangle_area = 0.5 * base * height

  # The area of the hexagon is 6 such triangles
  triangle_area * 6
end

def circle
  radius = 1
  PI * radius ** 2
end

puts "diamond == #{sprintf "%2.2f", (100 * diamond / circle)}%"
puts "hexagon == #{sprintf "%2.2f", (100 * hexagon / circle)}%"

And

$ ./geometrons.rb 
diamond == 63.66%
hexagon == 82.70%

Further, regular hexagons are highest-vertex polygon that form a regular tessellation of the plane.