You are here: HomeArticles → Mathemagic

# Mental math "mathemagic" with Arthur Benjamin (video)

This is an amazing performance of "mathemagic" – calculating things in one's head by Arthur Benjamin. He not only can calculate but also ENTERTAIN, so this video is sure to captivate you! I enjoyed it a lot, and my children watched it multiple times in a row.:)

Benjamin first squares a 2-digit number in his head – faster than the audience members with calculators can do it. He takes it a step further, and squares a 3-digit number in his head, again racing against calculators! And... he takes it a step further yet!

Benjamin also figures out what day of the week the birthdays of some audience members were and performs a few other tricks.

I encourage you to watch the whole video – you'll enjoy it!

The most interesting part might be the very end, where he takes on the challenge of squaring a five-digit number – AND explains how he does it!

And it's not "magic" as such; it's based on solid mathematical principles that we all use when multiplying numbers on paper.

To square any number n, he breaks it into two parts (a + b) and then uses this principle: (a + b)2 = a2 + 2ab + b2, which is all too familiar to algebra students.  To keep the partial results in his mind, he uses WORDS to represent numbers and memorizes the words instead of the partial results. I found that quite interesting and clever!

As an example with a 3-digit number (Benjamin didn't do this one), if you want to square 387, think of it as 300 + 87. Then, 3872 = (300 + 87)2 = 3002 + 2 · 300 · 87 + 872.

Now, 3002 is easy; it's 90,000. And 2 · 300 · 87 = 600 · 87 can be calculated as 6 · 87 and then you tag two zeros to it. 6 · 87 = 6 · 80 and 6 · 7 = 480 + 42 = 522. So 2 · 300 · 87 is 52,200.

Lastly, you can calculate 872 using the same principle: (80 + 7)2 = 802 + 2 · 80 · 7 + 72. Or perhaps you'd like use this shortcut and write it as 87 · 87 = (87 + 3) · (87 − 3) + 32 =  · 84 + 9 = 7200 + 360 + 9 = 7569.

Once you have that, just add all the partial results. (I'll leave the rest of the details to you.)

You can also view a transcript of the video here.