project_euler.problem_587.sol1¶
Project Euler Problem 587: https://projecteuler.net/problem=587
A square is drawn around a circle as shown in the diagram below on the left. We shall call the blue shaded region the L-section. A line is drawn from the bottom left of the square to the top right as shown in the diagram on the right. We shall call the orange shaded region a concave triangle.
It should be clear that the concave triangle occupies exactly half of the L-section.
Two circles are placed next to each other horizontally, a rectangle is drawn around both circles, and a line is drawn from the bottom left to the top right as shown in the diagram below.
This time the concave triangle occupies approximately 36.46% of the L-section.
If n circles are placed next to each other horizontally, a rectangle is drawn around the n circles, and a line is drawn from the bottom left to the top right, then it can be shown that the least value of n for which the concave triangle occupies less than 10% of the L-section is n = 15.
What is the least value of n for which the concave triangle occupies less than 0.1% of the L-section?
Functions¶
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Returns integral of circle bottom arc y = 1 / 2 - sqrt(1 / 4 - (x - 1 / 2) ^ 2) |
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Returns area of concave triangle |
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Returns least value of n |
Module Contents¶
- project_euler.problem_587.sol1.circle_bottom_arc_integral(point: float) float¶
Returns integral of circle bottom arc y = 1 / 2 - sqrt(1 / 4 - (x - 1 / 2) ^ 2)
>>> circle_bottom_arc_integral(0) 0.39269908169872414
>>> circle_bottom_arc_integral(1 / 2) 0.44634954084936207
>>> circle_bottom_arc_integral(1) 0.5
- project_euler.problem_587.sol1.concave_triangle_area(circles_number: int) float¶
Returns area of concave triangle
>>> concave_triangle_area(1) 0.026825229575318944
>>> concave_triangle_area(2) 0.01956236140083944
- project_euler.problem_587.sol1.solution(fraction: float = 1 / 1000) int¶
Returns least value of n for which the concave triangle occupies less than fraction of the L-section
>>> solution(1 / 10) 15