Problem61.py

# Code by @AmirMotefaker

# projecteuler.net
# https://projecteuler.net/problem=61

# Cyclical figurate numbers
# Problem 61

# 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.

import itertools


# Build table of numbers
# numbers[i][j] is the set of figurate numbers of i sides (3 <= i <= 8), having 4 digits, beginning with the 2 digits equal to j
def Calculate():
	numbers = [[set() for j in range(100)] for i in range(9)]
	for sides in range(3, 9):
		for n in itertools.count(1):
			num = figurate_number(sides, n)
			if num >= 10000:
				break
			if num >= 1000:
				numbers[sides][num // 100].add(num)
	
	def find_solution_sum(begin, current, bit_set, sum):
		if bit_set == 0b111111000:    
			if current % 100 == begin // 100:
				return sum
		else:
			for sides in range(4, 9):
				if (bit_set >> sides) & 1 != 0:
					continue
				for num in numbers[sides][current % 100]:
					temp = find_solution_sum(begin, num, bit_set | (1 << sides), sum + num)
					if temp is not None:
						return temp
			return None
	
	# Search
	for i in range(10, 100):
		for num in numbers[3][i]:
			temp = find_solution_sum(num, num, 1 << 3, num)
			if temp is not None:
				return str(temp)
	raise AssertionError("No solution")


def figurate_number(sides, n):
	return n * ((sides - 2) * n - (sides - 4)) // 2


if __name__ == "__main__":
	print(Calculate())