2016 AMC (Sample Question)

Let $k$ be a positive integer. Bernado and Silvia take turns writing and erasing numbers on a blackboard as follows: Bernardo starts by writing the smallest perfect square with $k+1$ digits. Every time Bernardo writes a number, Silvia erases the last $k$ digits of it. Bernardo then writes the next perfect square, Silvia erases the last $k$ digits of it, and this process continues until the last two numbers that remain on the board differ by at least 2. Let $f(k)$ be the smallest positive integer not written on the board. For example, if $k=1$ , then the numbers that Bernardo writes are $16,25,36,49,64$ , and the numbers showing on the board after Silvia erases are $1,2,3,4,5,6$ , and thus $f(1) = 5$ . What is the sum of digits of $f(2) + f(4) + f(6) + \cdots + f(2016)$ ?