Prior Knowlege can be harmful

Questions that arouse your own interests can be dangerous. This is because they can sometimes cloud your judgment or make you feel like you know something that isn’t explicitly stated in the text . While this may be true in the real world, don’t forget that you can’t bring any outside knowledge with you to the CAT.

The reason behind this is simple: anybody should be able to solve the question with the information provided in the question. Yes, you might already know something pertinent to the situation, but you cannot use it to solve the question unless it’s explicitly stated in the question. Especially on Critical Reasoning questions, these red herrings can come influence your decision without you even noticing it.

This doesn’t mean that you can’t get excited when a question mentions your favorite team; it just means that you have to maintain your objectivity regardless.

There are 16 teams in a hockey league and each team plays each of the others once. Given that each game is played by two teams, how many total games will be played?

A) 120
B) 169
C) 196
D) 230
E) 256

Now, ignoring that most leagues don’t play perfect round-robin tournaments because they are time consuming, but this question could be adopted to any sport of choice (perhaps even WWE wrestling) and would be solved the same way. You can enjoy the casual mention of hockey in this problem, but you’re free to imagine your favorite sport instead if it makes seeing the pattern easier for you.

Let’s approach this in a brute strength manner first and refine our strategy as we go along. Each team will have to play each other team in the league. This means that the first team, which we’ll call team 1 for simplicity, has to play against team 2, team 3, team 4, etc up until team 16. This would comprise of 15 matches for team 1. Next, we consider team 2. Team 2 already faced team 1, so that game is off the books, and their schedule would start against team 3, then team 4, etc, up until team 16. This would lead to 14 separate matches.

We seem to have something of a pattern here, but let’s do a third team just to compare our hypothesis (H0: It will be 13 matches. HA: We’ll have to find another way). Team 3 has already faced teams 1 and 2, meaning that their schedule begins at team 4, and then goes on to team 5, etc up until team 16. This does indeed add up to 13 more games being played. The pattern seems to hold up logically, every team plays one fewer game than the last because they’ve already faced any opponent with a team number lower than theirs.

Now, this approach gives the correct answer, but yields a difficult sequence to be summed: 15+14+13+12+11+10+9+8+7+6+5+4+3+2+1. We can shortcut this calculation because the sequence is comprised of consecutive integers, which means the total will be the average multiplied by the number of terms. Since the terms run from 1 to 15 (easier to see this forwards than backwards), the average is (1+15)/2 or 8, and there are 15 terms. 15 x 8 is 120, answer choice A, and this is the correct answer.

The brute force approach is rarely the best strategy, but it’s worth noting that it does get you to the correct answer. You can also shortcut this calculation by ignoring the fact that some teams have already played against one another in your initial count. That is to say: Team 1 has to face 15 opponents, and Team 2 has to face 15 opponents as well. Team 3 will end up facing 15 opponents too, and eventually all 16 teams will face 15 opponents, meaning the total number of games should be 15*16. This math isn’t trivial, but you can get to 240 relatively quickly. The problem with 240 is that you have double counted all the games (i.e. 1 vs. 2 and 2 vs. 1). Simply taking this product and dividing it by two will eliminate the double counting and yield the correct answer of 120.

The final strategy is that we’re essentially making all the unordered pairs of a group. This means we can use combinations to get the correct number. If we have n = 16 teams, and we’re trying to make all the combinations of 2 teams (k = 2), then we have a combination of the form:

n! / (k! * (n-k)!)

This formula gives us 16! / (2! * (16-2)!).

Solving for the subtraction gives us:

16! / (2! * 14!)

Simplifying by eliminating the redundant 14! from both numerator and denominator gives:

16 * 15 / 2.

This of course simplifies to 8 * 15 or the aforementioned 120. No matter the approach, you should get the same result, which is still choice A.

The CAT will ask you all kinds of questions about topics you’ve never heard of, but sometimes it will contain a topic that’s near and dear to your heart. It’s okay to be a little elated; you need some positive moments during the 3 hour CAT marathon. Just keep in mind that the question will be like any other problem, you solve it using the information contained in the question and your hours of CAT prep. If you do that properly, you’ll be able to put the puck in the net on test day.