Solve problem 587

python/e587.py

@@ -0,0 +1,61 @@
+from math import sqrt, pi, atan
+
+
+def get_intersection(n: int) -> tuple[int, int]:
+    """Compute the point where the line intersects the circle for a given
+    number of circles `n`.
+
+    Equation for circle: y = 1 - sqrt(2*x - x**2)
+    Equation for line: y = (1/n) * x
+    """
+
+    # Computed via Wolfram alpha; could be derived by setting the curves equal,
+    # isolating the root, and then applying the quadratic formula.
+    x_int = (-sqrt(2) * n ** (3 / 2) + n**2 + n) / (n**2 + 1)
+    y_int = x_int / n
+    return (x_int, y_int)
+
+
+def get_ratio_orange(n: int) -> float:
+    """Compute the ratio between the orange and the blue section. The blue
+    section is the L-section meaning the area outside of the quarter circle.
+    The orange area is the concave triangle created by the diagonal line."""
+
+    x, y = get_intersection(n)
+    area_blue = 1 - (pi / 4)
+
+    theta = atan((1 - x) / (1 - y))  # angle of concave triangle circle part
+    area_sector = theta / 2  # area of sector of concave triangle circle part
+
+    # area_orange = area triangle + area_rectangle + area_triangle - area_sector
+    area_orange = (0.5 * x * y) + ((1 - x) * y) + (0.5 * (1 - x) * (1 - y))
+    area_orange -= area_sector
+    return area_orange / area_blue
+
+
+def get_less_than(percentage: float) -> int:
+    ratio = percentage / 100
+    n_min, n_max = 1, 10000
+
+    while n_max - n_min > 1:
+        n_mid = (n_max + n_min) // 2
+        ratio_n = get_ratio_orange(n_mid)
+        if ratio_n < ratio:
+            n_max = n_mid
+        else:
+            n_min = n_mid
+
+    return n_max
+
+
+def euler_587():
+    assert get_ratio_orange(1) == 0.5
+    assert abs(get_ratio_orange(2) - 0.3646) <= 0.0001
+    assert get_less_than(10) == 15
+    return get_less_than(0.1)
+
+
+if __name__ == "__main__":
+    solution = euler_587()
+    print(f"e587.py: {solution}")
+    assert solution == 2240