Sometimes, we get questions which we cannot neatly bracket as Arithmetic/Algebra/Geometry etc. In fact, the higher level questions usually focus on more than one subject area. The trick in these questions is to assimilate all your knowledge from various areas and then think how best to solve.
Today we have such a question for you – you could get really lost in it or could solve it in a few seconds if you take the right track. The trick is starting on the right track and that is why you have 2 mins per question available to you else 40 secs per question would have been sufficient!
Question: This game season, five divisions are going to play. Out of all the teams in each division, 6, 9, 12, 13 and 14 teams have qualified from the respective divisions. Each division will hold its own tournament – where a team is eliminated from the tournament upon losing two games – in order to determine its champion. The five division champions will then play in a knock-off tournament – a team is eliminated as soon as it loses a game – in order to determine the overall champion. Assuming that there were no ties and no forfeits, what is the maximum number of games that could have been played in order to determine the overall league champion?
Solution: Is it a max-min problem? Perhaps, but which guiding principle of max-min will we use to solve this problem? First think on your own how you will solve this problem.
Will you focus on the method or the result i.e. will you worry about who plays against whom or just focus on each result which gives one loss and one win? If you don’t worry about the method and just focus on the result, you can use a concept of mixtures here. In mixture questions, we focus on one component and how it changes. Here, we need to keep track of losses. Let’s focus on those and forget about the wins. As given, there were no ties so every loss has a win on the other side.
Every time a game is played, someone loses. We can give at most 2 losses to a team since after that it is out of the tournament. Don’t worry about against who it plays those two games. Whenever a team loses 2 games, it is out. The team could have won many games but we are not counting the wins and hence, are not concerned about its wins. As discussed, we are counting the losses so each win of that team will be counted on the other side i.e. as a loss for the other team.
Consider the division which has 6 teams – what happens when 12 games are played? There are 12 losses and each team gets 2 losses (we can’t give more than 2 to a team since the team gets kicked out after 2 losses), so all are out of the tournament. But we need a winner so we play only 11 games so that the winning team gets only 1 loss. We want to maximize the losses (and hence the number of games), therefore the winning team must be given a loss too.
So maximum number of games that can be played by the district in its own tournament = 2*6 – 1 = 11
Similarly, the division with 9 teams can play at most 2*9 – 1 = 17 games.
The division with 12 teams can play at most 2*12 – 1 = 23 games.
The division with 13 teams can play at most 2*13 – 1 = 25 games.
The division with 14 teams can play at most 2*14 – 1 = 27 games.
This totals up to 11 + 17 + 23 + 25 + 27 = 103 games
Now we come to the games between the district champions.
We have 5 teams. 1 loss gets a team kicked out. If the teams play 4 games, there are 4 losses and 4 teams get kicked out. We have a final winner!
Hence the total number of games = 103 + 4 = 107
There are a lot of variations you can consider for this question.
Say, if we need to minimize the number of games, how many total games would have been played?
Notice that the only games you can avoid are the ones in which the 5 district champions lost. You do still need 2 losses for each team to get the district champion and one loss each for four district champions to get the winner. Hence, at least 107 – 5 = 102 games need to be played.
Look at it in another way:
To kick out a team, it needs to have 2 losses so if the district had 6 teams, there would be 5*2 = 10 games played.
Similarly, the division with 9 teams will play at least 2*8 = 16 games.
The division with 12 teams will play at least 2*11 = 22 games.
The division with 13 teams will play at least 2*12 = 24 games.
The division with 14 teams will play at least 2*13 = 26 games.
This totals up to 10 + 16 + 22 + 24 + 26 = 98 games
Now we have 5 champions and they will need to play at least 4 games to pick a winner.
Therefore, at least 98 + 4 = 102 games need to be played.
You can try other similar variations – what happens when a team is kicked out after it loses 3 games instead of 2? What happens if you don’t have the knock-off tournament and instead need each district champion to lose 2 games to get knocked out?
Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as this blog!