Round-robin tournament puzzle


In a round-robin Battleship tournament (everyone plays everyone once) two teams, A and C, had the same number of wins. Must there be three teams in the tournament, A, B, C, such that A beat B, B beat C, and C beat A? Note that in Battleship, it's impossible to tie.


Suppose that C beat A. If every team that beat C, also beat A, then C would have one more win than A. Therefore, there must be a team B, that beat C, but lost to A.


Both the problem and solution have been adapted slightly from a puzzle I found in "The Inquisitive Problem Solver."