Answer:


Complete Solution:

32
games. 
There
is more than one way to do this problem.
Consider
that every team except the winner loses exactly 1 game.
If there are 64 teams
in the tournament and 1 winner, then there were 63 losing
teams. This means there were 63 games.
If
there are 32 teams in the tournament and 1 winner, there
were 31 games. So,
63  31 = 32 games. 





Alternate Solution
One: 


Create
a table like the one below to determine the number of
games required for a 64team tournament:
Round 
No.
of Teams at Start of Round 
No.
of Games 
No.
of Winners 
1 
64 
32 
32 
2 
32 
16 
16 
3 
16 
8 
8 
4 
8 
4 
4 
5 
4 
2 
2 
6 
2 
1 
1 
Total
No. of Games 
63 
Add the numbers for the games in
the table to find the total number, 63, of games for 64
teams. A similar table can be used to find that a 32team
tournament requires 31 games. The difference is 63  31
= 32. 





Alternate Solution
Two: 


After
the first round of 32 games in 1999, the number of teams
was the same as in the 1985 tournament. So,
the difference in the number of games is the number of
games played in the first round, or 32. 





Alternate Solution
Three: 


Another
way is to consider the number of games necessary with
2 teams, then 3, and so on. If you can identify
a pattern, you can use that pattern to determine the number
of games for 32 and 64 teams:
#
of Teams 
1st
Round 
2nd
Round 
3rd
Round 
#
of Games 
2
teams
(A,B) 
A
vs. B 


1

3
teams
(A,B,C) 
A
vs. B
C does not play 
B
vs. C 

2

4
teams
(A,B,C,D) 
A
vs. B
C vs. D 
B
vs. D 

3

5
teams
(A,B,C,D,E) 
A
vs. B
C vs. D
E doesn't play 
B
vs. D
E still doesn't play 
B
vs. E 
4 
As shown in the table, in each
case, there is one less game than the number of teams.
A 64team tournament would require 63 games to determine
a winner, while a 32team tournament would require 31
games. The difference is 63  31 or 32 games. 





Alternate Solution
Four: 


A
graphical method to solve the problem uses tournament
brackets showing how the teams are scheduled to play.
Consider the brackets below:
With this set of brackets, you see that half of the teams
are grouped in pairs on the right and half on the left
to begin. The first round is played in the outside brackets
for a total of 16/2 or 8 games. The next round with teams
paired in the second set of brackets consists of 8/2 or
4 games. Continuing this process shows that for 16 teams,
there are 8 + 4 + 2 + 1, or 15 total games. Using brackets
can be done for any number of teams. For 64 teams, there
are 63 games; for 32 teams, 31 games. 63 – 31 = 32. 

