Problem: 1000 players compete in a tournament. In each round, players are matched with opponents, and the winner proceeds to the next round. If there are an odd number of players in a round, one player chosen at random sits out of that round. What is the total number of games are played in the tournament?
Solution: 999. Each player loses exactly one game, except for the winner of the tournament. Hence, in a tournament of players, there will be games played. More formally, we have a bijection between the losers of the tournament and the games played, where each loser is paired with the game he lost. Since there are losers and one winner, there are games played.
This is a wonderful argument, particularly because it is entirely based on logic. Another attempt might be combinatorical or number-theoretical, counting the number of games in each round. While this would work nice for tournaments where there are players, here it flounders. This method works much more generally, and stands as a testament to the power of logic.