Squaring Quickly

You may have seen the Fun Fact on squares ending in 5; Here's a trick that can help you square ANY number quickly.

It's based on the identity for the difference of squares, but with a twist! Can you figure it out?

542 = 50 * 58 + 42 = 2916. 
422 = 40 * 44 + 22 = 1764. 
372 = 34 * 40 + 32 = 1369.

You have to pretty proficient at multiplying one digit numbers by two digit numbers in your head to do this trick well. But if you master this, then you can build upon it in some amazing ways:

1162 = 100 * 132 + 162 = 13,200 + 256 = 13,456.

Thinking CREATIVELY about everything you learn, no matter how trivial it may seem, will allow you to find some really clever applications!

Presentation Suggestions:
If you practice this a LOT beforehand, you can start off by asking students to name any 2-digit number and you will do it in your head quickly. Then tell them the trick. But only do this with a LOT OF PRACTICE!

The Math Behind the Fact:
If you look closely, we are using the identity: 

a2 = (a-b)(a+b) + b2.

The reference contains more ideas for doing fast mental calculations. See also Fun Facts on .

How to Cite this Page: 
Su, Francis E., et al. “Squaring Quickly.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

References:
A. Benjamin and M. Shermer, Secrets of Mental Math, Three Rivers Press, 2006

Fun Fact suggested by:
Francis Su

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