Focus Changing

Consider the following question:

139 football teams compete in a tournament in which the team that loses a game leaves the tournament. If only one team can win the tournament, what is the total number of games that are played in the tournament?

One way to solve this question is to divide the games into rounds so that in the first round, one team is left out and 138 teams play 69 games, leaving 69 winners + the team left out = 70 teams for the next round. On the next round, the 70 remaining teams will play 35 more games, leaving 35 winners. In round three, 17 games (one team left out), 18 teams left. In round four, 9 games, 9 teams left. In round five, 4 games (one team left out) 5 team left. In round six, 2 games are played and 3 teams are left. In round seven, 1 game will be played, leaving 2 teams. In round 8, the last game will be played, leaving 1 winner. Now, add up all the games: 69+35+17+9+4+2+1+1=138 games.

This is a nice way to solve the question. Of course, it could be a lot harder -if there were 281 teams, for example.

There is, however, a better way: Consider that only one team can win. Hence, 138 teams have to be disqualified. Since in every game, one team is disqualified, 138 games are needed. That’s all. If there were 281 teams, 280 games were needed.

All we did was change the focus from the number of teams remaining after every game, to the number of teams that need to be disqualified. Try to use this approach in questions that you feel will require a lot of test time to solve. If it does not work for you, leave it and just solve the long way if it is feasible.
This technique is very useful, and should be practiced during the period.

Another example to illustrate change of focus:

How many different ways are there to assign three students to two classes so that each student will be assigned to the first class, the second class, or to none of the classes?

6, 8, 12, 18, 27
Take a few moments to try to solve it yourself before you continue reading.

One way to solve it is to start counting the different ways to assign the students and calculate the number of different options for each of these assignments. Try it yourself to find that this way will probably take over 5 minutes to complete.

The shorter way to solve this question is by changing the focus from the number of different options for each arrangement to the number of options each student has. Each student has 3 options: class 1, class 2, or none. Thus, there are 3x3x3=27 different ways. Again, the shift in focus helped us solve the question for us in no time.