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60 lines
1.9 KiB
Python
60 lines
1.9 KiB
Python
"""
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Project Euler Problem 75: https://projecteuler.net/problem=75
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It turns out that 12 cm is the smallest length of wire that can be bent to form an
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integer sided right angle triangle in exactly one way, but there are many more examples.
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12 cm: (3,4,5)
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24 cm: (6,8,10)
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30 cm: (5,12,13)
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36 cm: (9,12,15)
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40 cm: (8,15,17)
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48 cm: (12,16,20)
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In contrast, some lengths of wire, like 20 cm, cannot be bent to form an integer sided
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right angle triangle, and other lengths allow more than one solution to be found; for
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example, using 120 cm it is possible to form exactly three different integer sided
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right angle triangles.
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120 cm: (30,40,50), (20,48,52), (24,45,51)
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Given that L is the length of the wire, for how many values of L ≤ 1,500,000 can
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exactly one integer sided right angle triangle be formed?
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Solution: we generate all pythagorean triples using Euclid's formula and
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keep track of the frequencies of the perimeters.
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Reference: https://en.wikipedia.org/wiki/Pythagorean_triple#Generating_a_triple
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"""
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from collections import defaultdict
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from math import gcd
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def solution(limit: int = 1500000) -> int:
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"""
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Return the number of values of L <= limit such that a wire of length L can be
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formmed into an integer sided right angle triangle in exactly one way.
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>>> solution(50)
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6
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>>> solution(1000)
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112
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>>> solution(50000)
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5502
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"""
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frequencies: defaultdict = defaultdict(int)
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euclid_m = 2
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while 2 * euclid_m * (euclid_m + 1) <= limit:
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for euclid_n in range((euclid_m % 2) + 1, euclid_m, 2):
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if gcd(euclid_m, euclid_n) > 1:
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continue
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primitive_perimeter = 2 * euclid_m * (euclid_m + euclid_n)
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for perimeter in range(primitive_perimeter, limit + 1, primitive_perimeter):
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frequencies[perimeter] += 1
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euclid_m += 1
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return sum(1 for frequency in frequencies.values() if frequency == 1)
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if __name__ == "__main__":
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print(f"{solution() = }")
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