Arthur Benjamin’s Mathemagic

Arthur Benjamin calls himself a mathemagician.  His mathematical tricks are truly fascinating, and you can experience them for yourself through various videos.  He has done a couple of TED talks, including this one.

Arthur Benjamin: A Performance of “Mathemagic”

More recently, he stopped by Huffington Post to give a peek into some of the tricks in his new book The Magic of Math.

How To Square Any Number in Less than a Second, Without a Calculator

As I watched his “explanation” of how he squares the numbers, I was not completely satisfied.  Sure, it’s cool that a magician revealed his secret, but I wanted to know the why and not just the how.  So, I used a little algebra.

Arthur started his explanation with the number 12.  He went down to an easy number, 10.  Since he had to go down 2 to get from 12 to 10, he then went up 2 to go from 12 to 14.  He multiplied 10 and 14.  Then, he added the square of the amount he had to go up and down.  So, that’s 4.  The result was the answer.  He next did the process with 97.  Sure enough, it worked!  But, how?  And does it work with any real number?

I did a little algebra and now see the why, and it does work with any real number.

Let x = the number you are squaring.  Let n = amount you have to go up and down from x.  You multiply (x + n) and (x – n).  Many of you likely recognize the difference of squares.  x^2 – n^2.  Now, he added the square of that difference or n^2.  Well, this gives us x^2 – n^2 + n^2 = x^2.  This is exactly what we are trying to find!

Often, these mathematical parlor tricks can be seen with some variables and a little bit of algebra.  I hope that I didn’t disappoint anyone by revealing the secret, but I am not a magician and figure it is okay.


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