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 64-team 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 32-team 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 64-team tournament would require 63 games to determine a winner, while a 32-team 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.

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