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 127 matches are totally played, then 128 players participate in the tournament.
2. If 114 players participate in the tournament, then 113 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!!!