Pat Ballew posted these at Pat'sBlog, and I worked on them last night when I had trouble sleeping. They come from the puzzle corner of the Gazette of the Australian Mathematical Society.
Problem 1. Digital deduction.
The numbers 2^2009 and 5^2009 are written out on a piece of paper in the usual decimal notation. How many digits are on this piece of paper?
Problem 2. Piles of stones.
There are 25 stones sitting in a pile next to a blackboard. You are allowed to take a pile and divide it into two smaller piles of size a and b, but then you must write the number a×b on the blackboard. You continue to do this until you are left with 25 piles, each with one stone. What is the maximum possible sum of the numbers written on the blackboard?
A few of us worked them out in the comments over there. I won't spoil your fun if you're seeing it first here. What I liked about these problems was that from each of them I learned something new that extends well beyond the problem at hand.