Stupid logic problem: At the state basketball tournament, 42 basketball games are played to determine the winner of the tournament. After each game, the loser is eliminated from the tournament. How many teams are in the tournament?
work backwards ....
final game: 2 teams , 1 game
semi's : 4 teams , 2 game, total = 3 games
quarters: 8 teams, 4 games, total = 7 games
round before: 16 teams, 8 games, total = 15 games
before that : 32 teams, 16 gaMES, total = 31 games
before that: 64 teams, 32 games, total = 63 games
In a loser-elimination tournament, there cannot be a total of 42 games, unless some teams had byes.
Data seems flawed.
e.g.
for 32 teams there would be 31 games,
for 64 teams there would be 63 games
notice the the total number of teams must be a power of 2 and the total number of games is 1 less than the number of teams.
Check my thinking.
e.g. look at the draw for the men's singles at the
current US Open Tennis tournament
(Broken Link Removed)
To find the number of teams in the tournament, we need to understand the elimination process. In every game, one team is eliminated. Therefore, after the first game, 1 team is eliminated, leaving (n-1) teams, where 'n' is the initial number of teams in the tournament.
Since there are 42 games in total, we can set up the following equation:
(n-1) + (n-2) + (n-3) + ... + 1 = 42
This equation represents the sum of the number of remaining teams after each game. Now we need to solve for 'n'.
To simplify the equation, we can rewrite it as the sum of consecutive integers:
1 + 2 + 3 + ... + (n-2) + (n-1) = 42
To solve this equation, we can use the formula for the sum of consecutive integers, which is given by:
Sum = (n/2) * (first term + last term)
In this case, the first term is 1 and the last term is (n-1). Plugging these into the formula, we get:
42 = (n/2) * (1 + n - 1)
Simplifying further:
42 = (n/2) * n
Now we have a quadratic equation. Let's rearrange it:
n^2/2 = 42
Multiplying both sides by 2, we get:
n^2 = 84
Taking the square root of both sides:
n ≈ 9.17
Since the number of teams must be a whole number, we round up to the nearest whole number, giving us:
n = 10
Therefore, there are 10 teams in the tournament.