Source code in the ruby programming language

# 1. Divide n by 12. Remember the remainder (n is 8 for the eight queens
#    puzzle).
# 2. Write a list of the even numbers from 2 to n in order.
# 3. If the remainder is 3 or 9, move 2 to the end of the list.
# 4. Append the odd numbers from 1 to n in order, but, if the remainder is 8,
#    switch pairs (i.e. 3, 1, 7, 5, 11, 9, …).
# 5. If the remainder is 2, switch the places of 1 and 3, then move 5 to the
#    end of the list.
# 6. If the remainder is 3 or 9, move 1 and 3 to the end of the list.
# 7. Place the first-column queen in the row with the first number in the
#    list, place the second-column queen in the row with the second number in
#    the list, etc.


def n_queens(n)
  if n == 1
    return "Q"
  elsif n < 4
    puts "no solutions for n=#{n}"
    return ""
  end
 
  evens = (2..n).step(2).to_a
  odds = (1..n).step(2).to_a
 
  rem = n % 12  # (1)
  nums = evens  # (2)
 
  nums.rotate if rem == 3 or rem == 9  # (3)
 
  # (4)
  if rem == 8
    odds = odds.each_slice(2).flat_map(&:reverse)
  end
  nums.concat(odds)
 
  # (5)
  if rem == 2
    nums[nums.index(1)], nums[nums.index(3)] = nums[nums.index(3)], nums[nums.index(1)]
    nums << nums.delete(5)
  end
 
  # (6)
  if rem == 3 or rem == 9
    nums << nums.delete(1)
    nums << nums.delete(3)
  end
 
  # (7)
  nums.map do |q|
    a = Array.new(n,".")
    a[q-1] = "Q"
    a*(" ")
  end
end
 
(1 .. 15).each {|n| puts "n=#{n}"; puts n_queens(n); puts}

class Queen
  attr_reader :count
  
  def initialize(num=8, out=true)
    @num   = num
    @out   = out
    @row   = *0...@num
    @frame = "+-" + "--" * @num + "+"
    @count = 0
    add = Array.new(2 * @num - 1, true)       # \ direction check
    sub = Array.new(2 * @num - 1, true)       # / direction check
    solve([], add, sub)
  end
  
  private
  def solve(row, add, sub)
    y = row.size
    if y == @num
      print_out(row) if @out
      @count += 1
    else
      (@row-row).each do |x|
        next unless add[x+y] and sub[x-y]
        add[x+y] = sub[x-y] = false
        solve(row+[x], add, sub)
        add[x+y] = sub[x-y] = true
      end
    end
  end
  
  def print_out(row)
    puts @frame
    row.each do |i|
      line = @num.times.map {|j| j==i ? "Q " : ". "}.join
      puts "| #{line}|"
    end
    puts @frame
  end
end

(1..6).each do |n|
  puzzle = Queen.new(n)
  puts " #{n} Queen : #{puzzle.count}"
end

(7..12).each do |n|
  puzzle = Queen.new(n, false)                # do not display
  puts " #{n} Queen : #{puzzle.count}"
end