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Tennis Championship

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A tennis championship is played on a knock-out basis, i.e., a player is out of the tournament when he loses a match.

1. How many players participate in the tournament if 15 matches are totally played?
Answer: 16

2. How many matches are played in the tournament if 90 players totally participate?
Answer: 89


The matches in the championship may be visualized (starting backwards from the finals) as follows:

Match Total Matches Total Players
Finals 1 2
Semi Finals 1 + 2 = 3 4
Quarter Finals 1 + 2 + 4 = 7 8
Pre-Quarter Finals 1 + 2 + 4 + 8 = 15 16

Note that half the players lose their matches at each stage and are out of the tournament.

More importantly, the above table shows that the total number of matches played is always one less than the number of players participating in the tournament. This is basically because all players except the champion have to lose one match.

One can now easily answer the two questions posed.

1. If 15 matches are totally played, then 16 players participate in the tournament.

2. If 90 players participate in the tournament, then 89 matches are totally played.

Food for thought:

How would you organize such a tennis tournament for an arbitrary number of players (say, 19)? Do all players participate in the first round? Think it over!!!

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