Calculating Perfect Squares on the GMAT

The GMAT is an exam that evaluates how you think. The test is designed to measure your reasoning skills and gauge how successful you will be in business school. This means that the test is not simply trying to ascertain how much you already know. This is similar to the mantra of “Give a man a fish and you feed him for a day; teach a man to fish and you feed him for a lifetime”. If you happen to already know that 144 is 12^2, then any question that asks about this specific number becomes much easier. However, if the exam starts asking about 13^2 or 14^2, and you only know 12^2, then you must find some method to take your knowledge and apply it to new and unscripted problems.

The major difference between the GMAT and high school exams is that the questions are unpredictable. In high school, we’re taught to memorize certain pieces of information, and then regurgitate them on the final exam. If the question on the exam differed even slightly from the question we’ve committed to memory, we tended to panic, guess and generally fail to see the relationship between what we learned and what we were being asked to solve. As an example, if you know 12^2, you’re already 85.2% of the way to solving 13^2 (you know, roughly…). There is a fairly simple way to go from one perfect square to another, but before we talk about the general case solution, let’s go back to the beginning.

This pattern holds with 0^2, but for simplicity’s sake, let’s starts with 1^2. 1^2 expanded is 1 x 1, and this gives us a product of 1. Let’s look at the next perfect square: 2^2. 2 x 2 = 4. This is an increase of 3 from the first perfect square. The next perfect square is 3^2. 3 x 3 = 9. This represents an increase of 5 from the previous perfect square. Let’s do one more to cement the pattern: 4^2. 4 x 4 = 16. From the previous perfect square, we’ve increased by 7. The next perfect squares are 25, 36 and 49, representing subsequent increases of 9, 11 and 13, respectively. Indeed, each increase between subsequent perfect squares is a positive odd integer, and they’re in sequential order. It turns out that this pattern holds for all perfect squares, allowing us a shortcut to calculate them quickly. Let’s look at why this pattern holds.

From the initial perfect square of 1^2, we increase to 2^2. Consider this in two parts. We start with 1 x 1, and then we go to 1 x 2, and then finally to 2 x 2. What happens at each step? The first step brings up the total by 1, as we are adding another one of the initial element. The second step brings us up by 2, as we are adding another one of the new (n+1) element. This difference is what makes the perfect square 2^2 increase by (1+2=) 3 from the previous perfect square 1^2. Similarly, going from 2^2 to 3^2 can stop by the intermediary step of 2 x 3, and then 3 x 3. The first increase is of 2, and the second is of 3, giving a total of 5. For the general case, we can see that n^2 becomes (n+1) ^2 if we simply take n^2 and increase it by n and then increase it by n+1.

While this level of mathematics is not required on the GMAT, it certainly makes certain calculations much faster. Let’s return to our initial example of 12^2. Most (non-GMAT aficionado) people don’t know 13^2 offhand, but since elementary school has indoctrinated us with the multiplication table up to 12, the majority of people can easily recall that 12^2 is 144. Using this shortcut, we can see that 13^2 is 144 + 12 + 13. Adding these together, we get 169, the correct answer. 14^2 will similarly be 169 + 13 + 14, or 196, and so on.

I don’t consider this strategy a trick in any way, but rather a result of deeply understanding mathematical properties. This is the type of skill that’s rewarded on the GMAT, and it’s often rewarded by solving questions like this in under 2 minutes:

225, or 15^2, is the first perfect square that begins with two 2s. What is the sum of the digits of the next perfect square to begin with two 2s.
(A) 7
(B) 9
(C) 13
(D) 16
(E) 21

This is the type of question that could easily take 5 minutes on the GMAT. We have very little information, only that the number we want is a perfect square that begins with two 2s. Also, that it’s not 225, which is one a lot of people might know (especially if you live in a country with 15% sales tax). Even with a calculator, this question isn’t particularly trivial, so we’ll need to devise a strategy before randomly squaring numbers and hoping they begin with 22…

First things first, the next perfect square cannot possibly be 22x. The next perfect square after 15^2 is 16^2, which is 256 (you can get here any way you like). This means that the next perfect square has to be 22xx. This gives us an order of magnitude to shoot for. Until we have a better idea on which numbers to hone in on, let’s use easy numbers to get a sense of where we’re going:
20^2 = 400
30^2 = 900
40^2 = 1,600
50^2 = 2,500

Okay, so the number must be somewhere between 40 and 50. From here, it may be obvious that we need to be closer to 50, since 22xx is more than halfway between 1,600 and 2,500. As such, an astute test taker might try something like 47^2 or 48^2 and see how close they got. However, instead of guessing, let’s use our perfect square strategy to see how quickly we can calculate the correct number.

50^2 is 2,500. This means that if I were to calculate 49^2, I could simply take 2,500 and remove 50, then remove 49. This is the reverse of adding them together to get from 49^2 to 50^2. You can also think of this subtraction as 2,500 – 99, which means that 49^2 must be 2,401. A cursory test of the unit digit reveals that 9 x 9 would yield a unit digit of 1, so we’re on the right track. Similarly, going to 48^2 from 49^2 involves taking 2401 and subtracting 49 then 48. This would be 2,401 – 97, or 2,304. We got close to 22xx, but we’ll need one more step. 47^2 will be 48^2 – 47 – 48. This is equivalent to 2,304 – 95, leaving us with 2,209.

The number we need is a perfect square that begins with 22, so 2,209 is the correct term. From here, we need to add together the digits and get the total of 13, which is answer choice C.

While there is no direct method to answer questions such as these, it’s important to not use blind guessing, as this can waste a lot of time and won’t help you solve a similar question next time. Back solving is useless in a situation like this as well, so our options are somewhat limited. A simple strategy such as calculating signpost perfect squares like 30^2 and 40^2 is helpful, and in a case such as this can negate much of the difficulty of the problem. Since this exam is a test of how you think, don’t forget that any perfect square is just a hop, skip and a jump from the next perfect square.

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Ron Awad is a GMAT instructor for Veritas Prep based in Montreal, bringing you weekly advice for success on your exam.  After graduating from McGill and receiving his MBA from Concordia, Ron started teaching GMAT prep and his Veritas Prep students have given him rave reviews ever since.