TheAlgorithms-Python
59 строк · 1.9 Кб
1"""
2Project Euler Problem 75: https://projecteuler.net/problem=75
3
4It turns out that 12 cm is the smallest length of wire that can be bent to form an
5integer sided right angle triangle in exactly one way, but there are many more examples.
6
712 cm: (3,4,5)
824 cm: (6,8,10)
930 cm: (5,12,13)
1036 cm: (9,12,15)
1140 cm: (8,15,17)
1248 cm: (12,16,20)
13
14In contrast, some lengths of wire, like 20 cm, cannot be bent to form an integer sided
15right angle triangle, and other lengths allow more than one solution to be found; for
16example, using 120 cm it is possible to form exactly three different integer sided
17right angle triangles.
18
19120 cm: (30,40,50), (20,48,52), (24,45,51)
20
21Given that L is the length of the wire, for how many values of L ≤ 1,500,000 can
22exactly one integer sided right angle triangle be formed?
23
24Solution: we generate all pythagorean triples using Euclid's formula and
25keep track of the frequencies of the perimeters.
26
27Reference: https://en.wikipedia.org/wiki/Pythagorean_triple#Generating_a_triple
28"""
29
30from collections import defaultdict
31from math import gcd
32
33
34def solution(limit: int = 1500000) -> int:
35"""
36Return the number of values of L <= limit such that a wire of length L can be
37formmed into an integer sided right angle triangle in exactly one way.
38>>> solution(50)
396
40>>> solution(1000)
41112
42>>> solution(50000)
435502
44"""
45frequencies: defaultdict = defaultdict(int)
46euclid_m = 2
47while 2 * euclid_m * (euclid_m + 1) <= limit:
48for euclid_n in range((euclid_m % 2) + 1, euclid_m, 2):
49if gcd(euclid_m, euclid_n) > 1:
50continue
51primitive_perimeter = 2 * euclid_m * (euclid_m + euclid_n)
52for perimeter in range(primitive_perimeter, limit + 1, primitive_perimeter):
53frequencies[perimeter] += 1
54euclid_m += 1
55return sum(1 for frequency in frequencies.values() if frequency == 1)
56
57
58if __name__ == "__main__":
59print(f"{solution() = }")
60