Mental Math Shortcut: Perfect Squares – 1 | SAT Video Post

A Handy SAT Math Shortcut – N2 – 1

Many of us have a pretty good sense of what the squares of the first 15 integers are. Sure, you might be a bit shaky on ‘13’ and ‘14’ but you should be comfortable with the rest. Just to be sure, I’ll reproduce those below:

1^2 = 1
2^2 = 4
3^2 = 9
4^2 = 16
5^2 = 25
6^2 = 36
7^2 = 49
8^2 = 64
9^2 = 81
10^2 = 100

11^2 = 121
12^2 = 144
13^2 = 169
14^2 = 196
15^2 = 225

Here are more squares students tend to know:
16^2 = 256
20^2 = 400
25^2 = 625
30^2 = 900

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(If you know all these, that’s pretty solid! You don’t have to memorize anymore.)

Why did I even bring this up in the first place? Well, I have a cool mental math shortcut. Assuming you know the above, you also know the following:

11 x 13, 14 x 16, 15×17 and even the crazy 29×31.

How is that possible?

Well, what if I told you that n^2 – 1 = (n – 1)(n + 1)

Big deal, you say. You already know basic algebra? And what does this have to do with squares?

Well, let’s say n = 20.

See, by knowing that 20^2 = 400, then the product of one integer less than 20—the number 19—and one integer greater than 20—the number 21—will be 400-1, which equals 399.

Try it with any of the numbers above. For instance, we know that 12^2 = 144. Therefore, 11 x 13 = 143.

29 x 31?

Well, what’s 30^2 – 1.

Just like that, voila! You’ve doubled your knowledge of squares above.


  • Chris Lele

    Chris Lele is the Principal Curriculum Manager (and vocabulary wizard) at Magoosh. Chris graduated from UCLA with a BA in Psychology and has 20 years of experience in the test prep industry. He's been quoted as a subject expert in many publications, including US News, GMAC, and Business Because. In his time at Magoosh, Chris has taught countless students how to tackle the GRE, GMAT, SAT, ACT, MCAT (CARS), and LSAT exams with confidence. Some of his students have even gone on to get near-perfect scores. You can find Chris on YouTube, LinkedIn, Twitter and Facebook!

By the way, Magoosh can help you study for both the SAT and ACT exams. Click here to learn more!

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