Prove or disprove the existence of random variables X_n so that the expected value of X_n goes to infinity as n goes to infinity but X_n goes to 0 almost surely (maybe not as much of a problem, but a pretty useful thing to think about for people who use EV calculations to make decisions).

Find the number of ways to tile a 1000 by 1000 grid with white and black tiles so each tile is adjacent to exactly two tiles of the same color.

First is much easier than 2nd.

My University has a problem sheet question that presents the first problem in a nice way that feels more real-worldy