Once again we dip into official test questions for our monthly brain teaser! This one is an actual ACT question. If you struggle with it, take comfort in knowing that this one took me longer than I’d like to admit (I saw this question on an actual ACT that I took with high school students). There is a way to quickly narrow down and guess and there’s also a way to get to the answer through blunt trial and error. But, like all brain teasers, there’s an efficient and elegant way to solve this question, one that does not take 4 or 5 minutes! Give it a try!
A teacher assigns each of her 18 students a different integer from 1 through 18. The teacher forms pairs of study partners by using the rule that the sum of the pair of numbers is a perfect square. Assuming the 9 pairs of students follow this rule, the student assigned which number must be paired with the student assigned the number 1?
A. 16
B. 15
C. 9
D. 8
E. 3
Answer: B
As with many of our “brain teasers,” this one is an actual standardized test question (from the ACT). In fact, I saw this question on an actual ACT that I took (on a Saturday morning with high school students), and I was pretty shocked by how hard it was, especially because it was only question 51 out of 60 (ACT Math questions generally move in order of difficulty). And, a little embarrassingly, it took me probably 4 or 5 minutes to solve. In fact, I skipped the question after spending some time working on it, finished the rest of the test, and then came back to solve it at the end. Only in hindsight did I see the “clever” way to do it…again, I’m embarrassed since solving questions like this effectively is my job!
It’s pretty easy to see that the answer must be B, D, or E, since the sum must be a perfect square. But which one? That’s where it gets more difficult. Once you realize it must be 15, 8, or 3, you could pair the 1 with one of those three answers and then just start haphazardly pairing the remaining numbers together to verify that you can make all of the pairs sum to perfect squares. But that can take a long time, especially if you start with the wrong answer!
As with most standardized test questions and brain teasers alike, there’s bound to be a more efficient and “elegant” solution. Here the key is to realize that 16, 17, and 18 must be matched with numbers that will make the resulting pairs sum to 25, because the next perfect square, 36, is too high to reach. That means 16, 17, and 18 need to be paired with 9, 8, and 7, respectively.
From there, the task becomes a little easier because nearly half the numbers have been removed. If you pair 1 with 15, and then 2 with 14, you’ll start to see a pattern. You can essentially pair the remaining numbers from the “outside” and work your way in to the middle, summing all of the pairs to 16. So, 3 goes with 13, 4 with 12, 5 with 11, and 6 with 10. That’s it! So 1 must be paired with 15 and the answer is B!
Whew, that was hard!