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95 lines
2.6 KiB
Python
95 lines
2.6 KiB
Python
"""
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Project Euler Problem 587: https://projecteuler.net/problem=587
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A square is drawn around a circle as shown in the diagram below on the left.
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We shall call the blue shaded region the L-section.
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A line is drawn from the bottom left of the square to the top right
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as shown in the diagram on the right.
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We shall call the orange shaded region a concave triangle.
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It should be clear that the concave triangle occupies exactly half of the L-section.
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Two circles are placed next to each other horizontally,
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a rectangle is drawn around both circles, and
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a line is drawn from the bottom left to the top right as shown in the diagram below.
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This time the concave triangle occupies approximately 36.46% of the L-section.
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If n circles are placed next to each other horizontally,
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a rectangle is drawn around the n circles, and
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a line is drawn from the bottom left to the top right,
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then it can be shown that the least value of n
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for which the concave triangle occupies less than 10% of the L-section is n = 15.
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What is the least value of n
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for which the concave triangle occupies less than 0.1% of the L-section?
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"""
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from itertools import count
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from math import asin, pi, sqrt
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def circle_bottom_arc_integral(point: float) -> float:
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"""
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Returns integral of circle bottom arc y = 1 / 2 - sqrt(1 / 4 - (x - 1 / 2) ^ 2)
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>>> circle_bottom_arc_integral(0)
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0.39269908169872414
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>>> circle_bottom_arc_integral(1 / 2)
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0.44634954084936207
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>>> circle_bottom_arc_integral(1)
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0.5
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"""
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return (
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(1 - 2 * point) * sqrt(point - point**2) + 2 * point + asin(sqrt(1 - point))
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) / 4
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def concave_triangle_area(circles_number: int) -> float:
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"""
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Returns area of concave triangle
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>>> concave_triangle_area(1)
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0.026825229575318944
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>>> concave_triangle_area(2)
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0.01956236140083944
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"""
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intersection_y = (circles_number + 1 - sqrt(2 * circles_number)) / (
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2 * (circles_number**2 + 1)
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)
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intersection_x = circles_number * intersection_y
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triangle_area = intersection_x * intersection_y / 2
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concave_region_area = circle_bottom_arc_integral(
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1 / 2
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) - circle_bottom_arc_integral(intersection_x)
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return triangle_area + concave_region_area
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def solution(fraction: float = 1 / 1000) -> int:
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"""
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Returns least value of n
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for which the concave triangle occupies less than fraction of the L-section
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>>> solution(1 / 10)
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15
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"""
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l_section_area = (1 - pi / 4) / 4
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for n in count(1):
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if concave_triangle_area(n) / l_section_area < fraction:
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return n
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return -1
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if __name__ == "__main__":
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print(f"{solution() = }")
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