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* Added solution for Project Euler problem 91. Reference: #2695 * Added doctest for solution() in project_euler/problem_91/sol1.py * Update docstring and 0-padding in directory name. Reference: #3256 * Update sol1.py Co-authored-by: Dhruv <dhruvmanila@gmail.com>
60 lines
1.6 KiB
Python
60 lines
1.6 KiB
Python
"""
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Project Euler Problem 91: https://projecteuler.net/problem=91
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The points P (x1, y1) and Q (x2, y2) are plotted at integer coordinates and
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are joined to the origin, O(0,0), to form ΔOPQ.
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There are exactly fourteen triangles containing a right angle that can be formed
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when each coordinate lies between 0 and 2 inclusive; that is,
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0 ≤ x1, y1, x2, y2 ≤ 2.
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Given that 0 ≤ x1, y1, x2, y2 ≤ 50, how many right triangles can be formed?
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"""
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from itertools import combinations, product
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def is_right(x1: int, y1: int, x2: int, y2: int) -> bool:
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"""
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Check if the triangle described by P(x1,y1), Q(x2,y2) and O(0,0) is right-angled.
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Note: this doesn't check if P and Q are equal, but that's handled by the use of
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itertools.combinations in the solution function.
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>>> is_right(0, 1, 2, 0)
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True
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>>> is_right(1, 0, 2, 2)
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False
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"""
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if x1 == y1 == 0 or x2 == y2 == 0:
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return False
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a_square = x1 * x1 + y1 * y1
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b_square = x2 * x2 + y2 * y2
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c_square = (x1 - x2) * (x1 - x2) + (y1 - y2) * (y1 - y2)
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return (
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a_square + b_square == c_square
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or a_square + c_square == b_square
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or b_square + c_square == a_square
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)
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def solution(limit: int = 50) -> int:
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"""
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Return the number of right triangles OPQ that can be formed by two points P, Q
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which have both x- and y- coordinates between 0 and limit inclusive.
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>>> solution(2)
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14
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>>> solution(10)
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448
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"""
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return sum(
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1
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for pt1, pt2 in combinations(product(range(limit + 1), repeat=2), 2)
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if is_right(*pt1, *pt2)
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)
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if __name__ == "__main__":
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print(f"{solution() = }")
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