How many games are played in a knockout competition involving 2^n teams?

Question

How many additional games are needed to determine the second best team?

Please provide reasons

Answer 1

There are a lot of different ways these tournaments can be structured, but the general double-elimination (in which a loser can play for second place), involves (2^n-1)+(2^n-2) games. The first term, (2^n-1), is for the first place team, and the second term, (2^n-2), is for the second place team. Also note that I don't mean for example, in (2^n-1), that n-1 is the exponent; I mean 2^n, minus 1.

Steve

Answer 2

How many games are played in a knockout competition involving 2^n teams?

The first round involves 2^(n-1) games (half as many games as there are teams as there are 2 teams per game)

The next round 2^(n -2) games .... etc

So there are 2^(n - 1) + 2^(n - 2) + 2^(n - 3) + 4 + 2 + 1 games

Summing this GP with n terms and common ratio of 2 you get 2^n - 1 games.

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How many additional games are needed to determine the second best team?

To find the second best team you need to find the best of the losers so the teams knocked out need their own competition with the winners of the loser's round playing the teams knocked out in the next round

After the first round there are 2^(n-1) losers so their first round will produce 2^(n-2) winners who will play the 2^(n - 2) losers of the second round of winners and so forth until they have their own fina, where the winner of the losers pmlays the loser of the winners to ensure that second best team is found. This is necessary as the 2nd best team could be knocked out early by the team that eventually wins.

This means 2^(n -2) + 2^(n-3) + 2^(n - 4) + ... + 4 + 2 + 1 additional games ie 2^(n-1) - 1 additional games

Answer 3

assume that n = 1 so there is two teams and the number of games require to determine a champion would only be 1.

if n = 2, there are 4 teams, there would be 3 games played to determine a champion.

if n = 3, there are 8 teams, there would be 7 games to determine a champion.

if you'll follow the pattern, the number of games required to determine a champion is always 1 less than the number of teams involve.

so (2^n) - 1

there would be no additional games to determine the second best, because the second best team would be the one who loses at the final / championship game.

Answer 4

Other answerers have given the correct answer for the number of games needed to find a winner (i.e., 2^n - 1). I want to answer the second part of the question: how many more games are needed to find second best.

The basic assumption of a single-knockout tournament is that not only is a team better than a team it defeats, but it is better than the team that team has defeated. So if A defeats B, and B has defeated C, the assumption says that A is better than C.

When a team W has won the competition, it has played (n) games, and therefore defeated (n) other teams. Consider the team it defeated in the final (say, L). To get to the final, L played and won (n-1) games, and is therefore better than not only those teams, but all the teams they defeated, and so on. In fact, L is better (i.e., known to be better) than 2^(n-1) - 1 other teams.

This means that in order to find the second best team, we don't need to consider those teams -- in effect, L has already won the right to "represent" them in the competition for second best.

You can repeat this argument for each of the (n) teams that W beat on its way to the championship. As a group, they've defeated every team other than W, and no one other than W has defeated them, so they're the only possible candidates for second best.

Following genericman1998's elegant observation that k-1 games are needed to determine the best of k teams, the number of games needed to determine second best is (n-1).

There's one small catch: you can only start this kind of competition for second best after you know who wins the final.

Answer 5

Wal C answered already, but here is a short, simple way to think about it.

(2^n)-1 teams must get knocked out, in order to have a single, final champion team remaining at the end of the tournament.

For every game played, 1 team is knocked out: the losing team of that game.

Therefore, the number of games must equal the number of teams that are knocked out, (2^n)-1

Answer 6

=2^(n-1)+2^(n-2)+..........
+2^2+2^1+2^0
to determine second I don't
think any additional game is
required