feat: add Project Euler problem 587 solution 1 (#6269)

Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>

DIRECTORY.md

@@ -898,6 +898,8 @@
     * [Sol1](project_euler/problem_493/sol1.py)
   * Problem 551
     * [Sol1](project_euler/problem_551/sol1.py)
+  * Problem 587
+    * [Sol1](project_euler/problem_587/sol1.py)
   * Problem 686
     * [Sol1](project_euler/problem_686/sol1.py)
 

project_euler/problem_587/sol1.py

@@ -0,0 +1,94 @@
+"""
+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?
+"""
+
+from itertools import count
+from math import asin, pi, sqrt
+
+
+def 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
+    """
+
+    return (
+        (1 - 2 * point) * sqrt(point - point**2) + 2 * point + asin(sqrt(1 - point))
+    ) / 4
+
+
+def concave_triangle_area(circles_number: int) -> float:
+    """
+    Returns area of concave triangle
+
+    >>> concave_triangle_area(1)
+    0.026825229575318944
+
+    >>> concave_triangle_area(2)
+    0.01956236140083944
+    """
+
+    intersection_y = (circles_number + 1 - sqrt(2 * circles_number)) / (
+        2 * (circles_number**2 + 1)
+    )
+    intersection_x = circles_number * intersection_y
+
+    triangle_area = intersection_x * intersection_y / 2
+    concave_region_area = circle_bottom_arc_integral(
+        1 / 2
+    ) - circle_bottom_arc_integral(intersection_x)
+
+    return triangle_area + concave_region_area
+
+
+def 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
+    """
+
+    l_section_area = (1 - pi / 4) / 4
+
+    for n in count(1):
+        if concave_triangle_area(n) / l_section_area < fraction:
+            return n
+
+    return -1
+
+
+if __name__ == "__main__":
+    print(f"{solution() = }")