problem61.py

"""
Cyclical Figurate Numbers
Project Euler Problem #61
by Muaz Siddiqui

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(3n−1)/2      1, 5, 12, 22, 35, ...
Hexagonal       P6,n=n(2n−1)        1, 6, 15, 28, 45, ...
Heptagonal      P7,n=n(5n−3)/2      1, 7, 18, 34, 55, ...
Octagonal       P8,n=n(3n−2)        1, 8, 21, 40, 65, ...
The ordered set of three 4-digit numbers: 8128, 2882, 8281, has three interesting 
properties.

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).
Each polygonal type: triangle (P3,127=8128), square (P4,91=8281), and pentagonal 
(P5,44=2882), is represented by a different number in the set.
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.
"""
from euler_helpers import timeit

def figurate_nums(f, start, to):
    nums = []
    next = 0 
    n = 1
    while next < to:
        next = f(n)
        if next > start:
            nums.append(int(next))
        n += 1
    nums.pop()
    return nums

def remove_invalid(nums):
    return [num for num in nums if int(str(num)[-2:]) >= 10]

def triangular(start, to):
    return remove_invalid(figurate_nums(lambda x:x*(x+1)/2, start, to))

def square(start, to): 
    return remove_invalid(figurate_nums(lambda x:x**2, start, to))

def pentagonal(start, to):
    return remove_invalid(figurate_nums(lambda x:x*(3*x-1)/2, start, to))

def hexagonal(start, to):
    return remove_invalid(figurate_nums(lambda x:x*(2*x-1), start, to))

def heptagonal(start, to):
    return remove_invalid(figurate_nums(lambda x:x*(5*x-3)/2, start, to))

def octagonal(start, to):
    return remove_invalid(figurate_nums(lambda x:x*(3*x-2), start, to))

def begin(num):
    return int(str(num)[:2])

def end(num):
    return int(str(num)[-2:])

cycle = [0 for num in range(6)]
nums =[[] for num in range(6)]

def recursive_find(num, length):
    for a in range(len(cycle)):
        if cycle[a] != 0:
            continue
        for b in range(len(nums[a])):
            distinct = True
            for c in range(len(cycle)):
                if nums[a][b] == cycle[c]:
                    distinct = False
                    break;
            if distinct and begin(nums[a][b]) == end(cycle[num]):
                cycle[a] = nums[a][b]
                if length == 5:
                    if begin(cycle[5]) == end(cycle[a]):
                        return True
                if recursive_find(a, length + 1):
                    return True
        cycle[a] = 0
    return False 

@timeit
def answer(): 
    nums[0] = triangular(1000,10000)
    nums[1] = square(1000,10000)
    nums[2] = pentagonal(1000,10000)
    nums[3] = hexagonal(1000,10000)
    nums[4] = heptagonal(1000,10000)
    nums[5] = octagonal(1000,10000)

    for n in range(len(nums[5])):
        cycle[5] = nums[5][n]
        if recursive_find(5, 1):
            break
    print(cycle)
    return sum(cycle)