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Tennis tournament

Tennis tournament

A tennis tournament was organized a few years ago in my neighborhood club. 9 players signed up and the match diagram looked more or less like this:

As you will see, there were some difficulties with the organization due to the number of participants. The participant "I" went directly to the second round, the same happened with the winner between "A" and "B". In total eight matches were played. The following year, the championship was much more successful and there were a total of 135 registered.

Do you know how many games were played in total in the tournament?

The problem itself is not more difficult, it is a matter of taking the work of drawing the match diagram and counting the matches or of dividing by two and taking the remains to calculate the passes to another round ... but the grace is to find a Quick and easy way to respond without doing any calculation.

Solution

Since there are 135 participants and there can only be one champion, it can be deduced that 134 players must be eliminated. If in each game a player is eliminated 134 games must be played. Always a match less than the number of players since the champion does not need to be eliminated.