Stage Four
Tennis
Competition
In a sporting knockout competition the loser of each match drops out of the competition and does not play again.
If there are 32 tennis players in a knockout competition how many matches must be played to determine the winner?
At Wimbledon the best 128 tennis players in the world compete in a knockout competition. How many matches must be played to determine the winner?
If a local competition had 13 players, how may matches would need to be played. (Hint: draw up a table with different numbers of players in competition.)
Did you find a pattern?
Is there more than one pattern?
Tennis Competition - a solution
In a sporting knockout competition the loser of each match drops out of the competition and does not play again.
If there are 32 tennis players in a knockout competition how many matches must be played to determine the winner?
We'll work through this one and work out a pattern...
In the first round of the competition there will be 16 matches involving the 32 players.
In the second round there will be 8 matches involving the 16 winners of the first round.
In the third round there will be 4 matches involving the 8 winners of the second round.
In the fourth round there will be 2 matches involving the 4 winners of the third round.
In the fifth round there will 1 match involving the 2 winners of the fourth round
This gives a total of 16 + 8 + 4 + 2 + 1 or 31 matches to complete the competition.
At Wimbledon the best 128 tennis players in the world compete in a knockout competition. How many matches must be played to determine the winner?
Using the the pattern seen for 32 players:
64 + 32 + 16 + 8 + 4 + 2 + 1 = 127 games
There seems to be a pattern in that the number of games needed is 1 less than the number of players. Does this work in more complicated situations - with an uneven number of players? Lets try...
If a local competition had 13 players, how may matches would need to be played. (Hint: draw up a table with different numbers of players in competition.)
In the first round there can only be 6 games - one player has to miss out.
Therefore there will be 6 winners, plus the player who missed out for the second round.
In the second round there are 7 players but there can only be 3 games - again 1 player has to miss out.
There will be three winners, plus the player who missed out for the third round.
So, in the 3rd round there are 4 players and 2 games.
There are 2 winners to go into the last round
The last round has one game to give a winner.
6 + 3 + 2 + 1 = 12
Did you find a pattern?
Yes, the number of games needed seems to be 1 less than the total number of players.
Is there more than one pattern?
That was the only pattern that I found. Did you find another?