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 $$f(2) + f(4) + f(6) + \cdots + f(2016)$$?

×