Suppose that we have $32$ teams in a tournament, we divide them into $8$ groups, where each group has $4$ teams, in each group, every team will play two games with the rest $3$ teams and each game can be win/loss/tie, now I want to prove that there must exist two teams having same number of wins,losses and ties.
2026-04-03 14:53:13.1775227993
Prove that in a $32$ team games, there must exist $2$ teams with same number of wins,losses and ties.
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Hint:
Suppose for a particular team, there are $a$ wins, $b$ losses and $c$ ties, then we have $a+b+c=6$ (Why?).
Count the number of solutions to the above equation and compare them to the number of teams we have. This is what is known as pigeonhole principle.