## Problem of the Day #109: Recursive Functions Are Cool
*July 6, 2011*

*Posted by Alex in : potd , trackback*

Given the recursive function, $f(x) =$

$$

\begin{cases}

0 & \text{if } x \le 2 \\

1 + \frac{f(x – 3) + f(x – 2) + f(x – 1)}{3} & \text{if } x > 2 \text{ and } x \equiv 0 \pmod{3} \\

2 + \frac{6f(x – 3) + 3f(x – 2) – f(x – 1)}{8} & \text{if } x > 2 \text{ and } x \equiv 1 \pmod{3} \\

3 + \frac{f(x – 3) + 2f(x – 2) + f(x – 1)}{4} & \text{if } x > 2 \text{ and } x \equiv 2 \pmod{3}

\end{cases},

$$

find the nearest integer to $f(2222)$. Feel free to use a four-function calculator (nothing except for $+$, $-$, $\times$, and $\div$).

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