GMP: Gratuitous Math Post

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I love my math senior seminar a lot, and had an obscene amount of fun with one of the problems due next Tuesday.

7. In a round-robin tennis tournament with n players, each player plays every other player exactly once. Can one always rank the players in such a way that the player ranked 1st has beaten the player ranked 2nd, the player ranked 2nd has beaten the player ranked 3rd, …, the player ranked (n-1)st has beaten the player ranked nth?

Why yes, yes one can. Here’s how to do it:
-Step 1. If there are players to choose from, choose player p and proceed to step 2. If there are no players left to choose, proceed to step 5.
-Step 2. If there are no players listed in the rank list, give p rank 1 and repeat step 1. If there is at least one player on the rank list, let n=1 and proceed to step 3.
-Step 3. If p beat the person with rank n, give player with rank n the new rank n+1, the player with rank n+1 the new rank n+2, etc., give p rank n, and repeat step 1. If p did not beat the person with rank n, and therefore was beaten by the person with that rank, proceed to step 4.
-Step 4. If there is a player ranked n+1, repeat step 3, using n+1 in place of n. If there is no player ranked n+1, rank p n+1 and repeat step 1.
-Step 5. You’re done making your shiny new list of players ranked such that each player ranked (n-1)st has beaten the player ranked nth.