In which case, since 35 is odd one team at the beginning moves on without playing. (I forget what the correct term for that is. Is it a "bye" or something?) So the first round eliminates teams, leaving 18.
The second round eliminates 9 teams, leaving 9 teams.
The third round includes that "bye" thing (or whatever), so teams are eliminated, leaving 6 teams.
The fourth round eliminates 3 teams, leaving 3 teams.
The fifth round includes a bye, so team is eliminated leaving 2 teams.
The sixth round is the final and eliminates 1 team.
So the winning team plays 6 games.
I believe that a bye does technically count as a game, but you are right... they really didn't play did they? Then the answer to my question must vary between 3 and 6 games.
Since this isn't a definite answer I suppose that means my thought is pointless. Ah well!