# nayuki/Project-Euler-solutions

Switch branches/tags
Nothing to show
Fetching contributors…
Cannot retrieve contributors at this time
83 lines (66 sloc) 2.32 KB
 # # Solution to Project Euler problem 86 # Copyright (c) Project Nayuki. All rights reserved. # # https://www.nayuki.io/page/project-euler-solutions # https://github.com/nayuki/Project-Euler-solutions # import fractions, itertools def compute(): # solutions[k] is the set of all solutions where the largest side has length k. # A solution is a triple (x, y, z) such that 0 < x <= y <= z, and in the rectangular prism with dimensions x * y * z, # the shortest surface path from one vertex to the opposite vertex has an integral length. solutions = [] # Generates all solutions where the largest side has length less than 'limit'. def generate_solutions(): # Pythagorean triples theorem: # Every primitive Pythagorean triple with a odd and b even can be expressed as # a = st, b = (s^2-t^2)/2, c = (s^2+t^2)/2, where s > t > 0 are coprime odd integers. # Now generate all Pythagorean triples, including non-primitive ones. for s in itertools.count(3, 2): for t in range(s - 2, 0, -2): if s * s // 2 >= limit * 3: return if fractions.gcd(s, t) == 1: for k in itertools.count(1): a = s * t * k b = (s * s - t * t) // 2 * k c = (s * s + t * t) // 2 * k if a >= limit and b >= limit: break find_splits(a, b, c) find_splits(b, a, c) # Assumes that a^2 + b^2 = c^2. def find_splits(a, b, c): z = b for x in range(1, a): y = a - x if y < x: break if c * c == min( (x + y) * (x + y) + z * z, (y + z) * (y + z) + x * x, (z + x) * (z + x) + y * y): temp = max(x, y, z) if temp < limit: # Add canonical solution item = tuple(sorted((x, y, z))) solutions[temp].add(item) # cumulativesolutions[m] = len(solutions[0]) + len(solutions[1]) + ... + len(solutions[m]). cumulativesolutions = [0] limit = 1 while True: # Extend the solutions list with blank sets while len(solutions) < limit: solutions.append(set()) generate_solutions() # Compute the number of cumulative solutions up to and including a certain maximum size for i in range(len(cumulativesolutions), limit): sum = cumulativesolutions[i - 1] + len(solutions[i]) cumulativesolutions.append(sum) if sum > 1000000: return str(i) # Raise the limit and keep searching limit *= 2 if __name__ == "__main__": print(compute())