We present original solutions to four challenging mathematical puzzles. The first two are concerned with random processes. The first, here called The President's Welfare Plan, can be reduced to computing, for arbitrary large values of n, the expected number of iterations of a program that increases a variable at random between 1 and n until exceeds n. The second one, called The Dining Researchers, can be reduced to determining the probability of reaching a given point after visiting all the others in a circular random walk. The other two problems, called Students vs Professor and Students vs Professor II, involve finding optimal winning group strategies in guessing games.