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21 Nov 2022, 06:42
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AkamaiBrah
Posted from my mobile device
Please advise if we go by this approach then each team has played only against team 1 and 2 only and they didn't not play any matches with each other, thus all those teams played 2 matches each where as a and b together played 17 matches.
Doesn't this approach give us minimum number or matches and not maximum
21 Nov 2022, 17:57
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Hi puneetfitness,
Regardless of whether we wanted the MINIMUM number or the MAXIMUM number of games played, each team that was NOT the respective chamption of its division would have to lose 2 times - which would require 2 games for each of those teams. The chamption of each of the 4 divisions could potentially lose 1 game in its division and still be the champion of that division (so those 4 additional games) are what would maximize the number of games played. Either way, there would the be 3 more games (among the 4 division champions) to determine the overall league chamption.
Here's how to calculate the maximum number of games. In a double-elimination tournament, a team that loses 2 times is eliminated, so every team EXCEPT for the 'champion' will lose twice (and the champion will lose either 0 or 1 times). We're asked to MAXIMIZE the number of games played, so we'll need each champion to lose 1 time.
Double-elimination Division games:
9 teams = (8 teams)(2 losses each) + 1 loss for the champ = 17 games
10 teams = (9 teams)(2 losses each) + 1 loss for the champ = 19 games
11 teams = (10 teams)(2 losses each) + 1 loss for the champ = 21 games
12 teams = (11 teams)(2 losses each) + 1 loss for the champ = 23 games
In the single-elimination tournament for the 4 champions, 3 of the teams will lose once and the winner will lose 0 times. That will require 3 more games total.
Maximum total games: 17 + 19 + 21 + 23 + 3 = 83 total games
GMAT assassins aren't born, they're made,
Rich
Regardless of whether we wanted the MINIMUM number or the MAXIMUM number of games played, each team that was NOT the respective chamption of its division would have to lose 2 times - which would require 2 games for each of those teams. The chamption of each of the 4 divisions could potentially lose 1 game in its division and still be the champion of that division (so those 4 additional games) are what would maximize the number of games played. Either way, there would the be 3 more games (among the 4 division champions) to determine the overall league chamption.
Here's how to calculate the maximum number of games. In a double-elimination tournament, a team that loses 2 times is eliminated, so every team EXCEPT for the 'champion' will lose twice (and the champion will lose either 0 or 1 times). We're asked to MAXIMIZE the number of games played, so we'll need each champion to lose 1 time.
Double-elimination Division games:
9 teams = (8 teams)(2 losses each) + 1 loss for the champ = 17 games
10 teams = (9 teams)(2 losses each) + 1 loss for the champ = 19 games
11 teams = (10 teams)(2 losses each) + 1 loss for the champ = 21 games
12 teams = (11 teams)(2 losses each) + 1 loss for the champ = 23 games
In the single-elimination tournament for the 4 champions, 3 of the teams will lose once and the winner will lose 0 times. That will require 3 more games total.
Maximum total games: 17 + 19 + 21 + 23 + 3 = 83 total games
GMAT assassins aren't born, they're made,
Rich
21 Nov 2022, 22:35
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puneetfitness
Every match played gives a loss (and a win). Some team must register that loss. The moment a team has two losses, it is out. So for every game, a loss is going to go to some team.
Consider the division with 9 teams. What happens when 9 games have been played? There have been 9 losses. Any team can register at most 2 losses (after that it is out). We can either give one loss to each team and kick nobody out or we can allocate 2 losses to some teams and remove them or some combination thereof.
Say we give 1 loss to each team. Now what happens when the 10th match is played? A team has 2 losses and is out. What happens 11th match is played? Another team is out and so on...
Now if you see, at most 17 matches can be played.
