Triangle, square, pentagonal, hexagonal, heptagonal, and octagonal numbers are all figurate (polygonal) numbers and are generated by the following formulae:

Triangle		P3,n=n(n+1)/2		1, 3, 6, 10, 15, ...
Square		P4,n=n2		1, 4, 9, 16, 25, ...
Pentagonal		P5,n=n(3n1)/2		1, 5, 12, 22, 35, ...
Hexagonal		P6,n=n(2n1)		1, 6, 15, 28, 45, ...
Heptagonal		P7,n=n(5n3)/2		1, 7, 18, 34, 55, ...
Octagonal		P8,n=n(3n2)		1, 8, 21, 40, 65, ...

The ordered set of three 4-digit numbers: 8128, 2882, 8281, has three interesting properties.

1. The set is cyclic, in that the last two digits of each number is the first two digits of the next number (including the last number with the first).
2. Each polygonal type: triangle (P_3,127=8128), square (P_4,91=8281), and pentagonal (P_5,44=2882), is represented by a different number in the set.
3. This is the only set of 4-digit numbers with this property.

Find the sum of the only ordered set of six cyclic 4-digit numbers for which each polygonal type: triangle, square, pentagonal, hexagonal, heptagonal, and octagonal, is represented by a different number in the set.

+-#triangles
def triangles(n)
  (n*(n+1))/2
end

+-#squares
def squares(n)
  n*n
end

+-#pentagonals
def pentagonals(n)
  (n*(3*n-1))/2
end

+-#hexagonals
def hexagonals(n)
  n*(2*n-1)
end

+-#heptagonals
def heptagonals(n)
  (n*(5*n-3))/2
end

+-#octagonals
def octagonals(n)
  n*(3*n-2)
end

+-#Enumerable
module Enumerable
  def sum
    self.inject{|u,v|u+v}
  end
  def product
    self.inject{|u,v|u*v}
  end
  def count(ob)
    self.select{|i|i==ob}.length
  end
  def take_while
    return [] unless yield(self.first)
    if(self.class==Range)
      evalPoint=self.first
      evalPoint=evalPoint.succ while yield(evalPoint.succ) and not evalPoint==self.last
      return self.first..evalPoint
    else
      s=self.to_a
      upTo=1
      upTo+=1 while yield(s[upTo]) and upTo<self.length
      return self[0...upTo]
    end
  end
end

def binary_interval(list,range)
  range=range.begin..(range.end-1) if range.exclude_end?
  i=list.length/2
end

def p61recur(so_far,available)
  return [so_far] if(available.length==0)
  res=[]
  available.each do |k|
    possible=$p61polys[k]
    possible=possible.select{|i|i/100==so_far.last%100} if so_far.length>0
    possible=possible.select{|i|i%100==so_far.first/100} if so_far.length==5
    new_res=possible.map{|p|p61recur(so_far+[p],available-[k])}
    res+=new_res.sum if new_res.length > 0
  end
  res
end

def p61
  $p61polys=Hash.new([])
  i=1
  i+=1 until octagonals(i) >= 1000
  while(triangles(i)<10000)
    v=[triangles(i),squares(i),pentagonals(i),hexagonals(i),heptagonals(i),octagonals(i)]
    (0...6).each{|j|$p61polys[j]+=[v[j]] if v[j].between?(1000,9999)}
    i+=1
  end
  puts p61recur([],(0...6).to_a).first.sum
end