Fair soccer championship
The games played in the soccer world championship form a binary tree, where only the winner of each game moves up the tree (ignoring the initial games, where the teams are placed into groups of 4, 2 of which of which go onto play in the tree of games I just described). Assuming that the teams can be totally ordered in terms of how good they are, the winner of the championship will indeed be the best of all of the teams. However, the second best team does not necessarily get a second place in the championship. How many additional games need to be played in order to determine the second best team?
If we assume that each team wins its game if it better than the opponent team then:
We don’t know which is the 2nd best team but we do know that it could loose only to the best team but we don’t know on what round. So if arrange all the teams in a tree than if the tree level up to the winner is k
than the best team played k-1 games, and to determine the 2nd best only those teams that played against the best team should play between the in the same system so we need only k-2 more games.
For example: If we have 16 teams (1-16) and each odd number is the winner and the 2nd best team is 2.
Than on the 1st round #2 is eliminated.
on the 2nd round #3 is eliminated.
on the 3rd round #5 is eliminated.
on the 4th round #9 is eliminated and #1 is the winner. k=5 so #1 played 4 games. Now the eliminated teams who played against #1 are 2,3,5,9 so 2 will play against 3 and 5 against 9 and since #2 is the 2nd best team on the next round we get #2 against #5 (or #9)
so in total we had 3 more games (k=5-2)
Your Answer
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