I’m going to teach you how to multiply any two numbers between 1 and 100, in your head, as quickly as it takes to input them in a calculator. And it’s dead simple.

(It does require some memorization though)

So, I figured out this technique – probably somebody else figured it out before me, but I’ve never heard of it before – that reduces the task of multiplying any two numbers together into some addition, some subtraction, and one division by two. It’s pretty simple. In fact, it all just boils down to one equation:

If your eyes glazed over at the above equation, be strong! Do the FOIL method and check it out for yourself. This is about as hard as the math gets.

Cool! What does that equation mean, though? Well, any two numbers (we’ll call them ‘a’ and ‘b’) can be rewritten in terms of ‘x’ and ‘n’ —

That is, we say that n is half of the difference between our numbers a and b, and that x is the number halfway between a and b. Now, b = x – n and a = x+n! Therefore, for any a and b,

We’re going to run through these formulas using some example numbers, to keep you following along, before we talk about how to solve x^{2} – n^{2}. Let’s take the following two sets of numbers:

49 x 74

These numbers are big enough that most people wouldn’t even bother try to multiply them in their heads. Is it easier with this new method? (Yes.)

First, find n, half of the difference between the two numbers…

41 – 23 = 18

n = (18 / 2) = 9

74 x 49:

74 – 49 = 25

n = (25 / 2) = 12.5

So the halfway point between 23 and 41 is

And the halfway point between 49 and 74 is

This means that…

^{2}– 9

^{2}

49 x 74 = 61.5

^{2}– 12.5

^{2}

But what are 61.5^{2} , 12.5^{2} , 32^{2} and 9^{2}?

This is the hard part of the method. I don’t expect you to be able to solve 61.5^{2} in your head. Instead, you’re going to have to memorize 100 squares – the square of every integer from 0 to 100. Yeah.

It’s a lot of work, but it’s entirely doable. There’s only 100 numbers, and you probably already know the squares of, like, 30 of them. If you’re thinking of giving up now, just think of all the sexy mathematician ladies (or dudes) you’ll be picking up at parties once you’ve memorized these and can multiply in a second!

If sexy mathematician ladies (or dudes) aren’t enough to make you memorize all 100 squares, though, there are tricks you can use to calculate squares very quickly in your head. I’m not going to talk about them here. I’m not supporting your laziness. But if you’re interested, you should read up on Vedic Mathematics, it’s got some pretty cool stuff about mental algebra.

But seriously, go the hardcore route, memorize this table. If you do so, you’ll already know the squares of all those integers-plus-one-half like 61.5^{2} (but more on that later):

x | x^{2} |
x | x^{2} |
x | x^{2} |
x | x^{2} |
---|---|---|---|---|---|---|---|

0 | 0 | 25 | 625 | 50 | 2500 | 75 | 5625 |

1 | 1 | 26 | 676 | 51 | 2601 | 76 | 5776 |

2 | 4 | 27 | 729 | 52 | 2704 | 77 | 5929 |

3 | 9 | 28 | 784 | 53 | 2809 | 78 | 6084 |

4 | 16 | 29 | 841 | 54 | 2916 | 79 | 6241 |

5 | 25 | 30 | 900 | 55 | 3025 | 80 | 6400 |

6 | 36 | 31 | 961 | 56 | 3136 | 81 | 6561 |

7 | 49 | 32 | 1024 | 57 | 3249 | 82 | 6724 |

8 | 64 | 33 | 1089 | 58 | 3364 | 83 | 6889 |

9 | 81 | 34 | 1156 | 59 | 3481 | 84 | 7056 |

10 | 100 | 35 | 1225 | 60 | 3600 | 85 | 7225 |

11 | 121 | 36 | 1296 | 61 | 3721 | 86 | 7396 |

12 | 144 | 37 | 1369 | 62 | 3844 | 87 | 7569 |

13 | 169 | 38 | 1444 | 63 | 3969 | 88 | 7744 |

14 | 196 | 39 | 1521 | 64 | 4096 | 89 | 7921 |

15 | 225 | 40 | 1600 | 65 | 4225 | 90 | 8100 |

16 | 256 | 41 | 1681 | 66 | 4356 | 91 | 8281 |

17 | 289 | 42 | 1764 | 67 | 4489 | 92 | 8464 |

18 | 324 | 43 | 1849 | 68 | 4624 | 93 | 8649 |

19 | 361 | 44 | 1936 | 69 | 4761 | 94 | 8836 |

20 | 400 | 45 | 2025 | 70 | 4900 | 95 | 9025 |

21 | 441 | 46 | 2116 | 71 | 5041 | 96 | 9216 |

22 | 484 | 47 | 2209 | 72 | 5184 | 97 | 9409 |

23 | 529 | 48 | 2304 | 73 | 5329 | 98 | 9604 |

24 | 576 | 49 | 2401 | 74 | 5476 | 99 | 9801 |

Now, here’s a cool trick: even though we need to know integers-plus-one-half like 61.5 for this method, we already know them if we know the square of that number’s closest integer (rounded down). This is because:

^{2}

= (x+0.5)(x+0.5)

= x

^{2}+ x + 0.25

And for bonus points, we can ignore that “+ 0.25” completely! It’s just going to be subtracted out by the second half of our equation! Forget about it! Therefore…

^{2}= 61

^{2}+ 61 = 3721 + 61 = 3782

12.5

^{2}= 12

^{2}+ 12 = 144 + 12 = 156

Now, it should be easy for you solve those two problems.

^{2}– 9

^{2}= 1024 – 81 = 943

49 x 74 = 61.5

^{2}– 12.5

^{2}= (3721 + 61) – (144 + 12) = 3782 – 156 = 3626

And there you go! To recap this method:

- Find half of the difference between the two numbers you’re trying to multiply
- Add that to the smaller number to get the midpoint between your two numbers
- Midpoint squared minus half-of-difference squared equals your result!
- If the number is an integer-plus-one-half, remember that (x+0.5)
^{2}= x^{2}+ x (for our purposes)

There’s no reason you can’t use this method to multiply two numbers over 100, either! You just have to memorize your squares tables up to the highest number you’re willing to multiply.

I hope you get tons of use out of this and impress all the sexy mathematician ladies (or dudes)!

luisAnd that’s why I always use my phone!

luisThis trick it’s hard for me, but is a very good exercise to my brain.