In or Out

Born in Warsaw, Benoît Mandelbrot (1924-2010) is considered the father of fractal geometry. He studied mathematical processes that described self-similar and natural shapes known as fractals. Perhaps his most well-known contribution is the Mandelbrot set, which is pictured below (the set contains the black points):

The Mandelbrot set is typically drawn on the complex plane, a 2-dimensional plane representing all complex numbers. The horizontal axis represents the real portion of the number, and the vertical axis represents the imaginary portion. A complex number $c = x + yi$ (at position $(x, y)$ on the complex plane) is not in the Mandelbrot set if the following sequence diverges:

$z_{n+1} \leftarrow z_n^2 + c$

beginning with $z_0 = 0$. That is, $\lim_{n \to \infty}{|z_n|} = \infty$. If the sequence does not diverge, then $c$ is in the set.

Recall the following facts about imaginary numbers and their arithmetic:

$i = \sqrt{-1}$, $i^2 = −1$, $(x + yi)^2 = x^2 − y^2 + 2xyi$, $|x + yi| = \sqrt{x^2 + y^2}$

where $x$ and $y$ are real numbers, and $| \cdot |$ is known as the modulus of a complex number (in the complex plane, the modulus of $x + yi$ is equal to the straight-line distance from the origin to the the point $(x, y)$).

Write a program which determines if the sequence $z_n$ diverges for a given value $c$ within a fixed number of iterations. That is, is $c$ in the Mandelbrot set or not? To detect divergence, just check to see if $|z_n| > 2$ for any $z_n$ that we compute – if this happens, the sequence is guaranteed to diverge.

Each test case is described by a single line containing three numbers: two real numbers $-3 \le x \le 3$ and $-3 \le y \le 3$, and an integer $0 \le r \le 10\,000$. The value of $c$ for this case is $x + yi$, and $r$ is the maximum number of iterations to compute.

For each case, display the case number followed by whether the given $c$ is in the Mandelbrot set, using IN or OUT.