Imagine a round-robin chess tournament for 150 players, each of whom plays 7 games. (In other words, each player is guaranteed to participate in precisely 7 games with 7 different opponents. Remember...

Imagine a round-robin chess tournament for 150 players, each of whom plays 7 games. (In other words, each player is guaranteed to participate in precisely 7 games with 7 different opponents. Remember that each game has two players.)

Suppose that would-be draws in this tournament are resolved by a coin flip, so that every game has a winner and a loser. Prove that there must be two participants in such a tournament who have precisely the same sequence of wins and losses (for example, WWWLLLW).