2015-01-17

Many years ago, while lost in my thoughts, I discovered a new way to multiply a pair of numbers together in my head. It works well with certain pairs of numbers and is useless with others. Which pairs will work depends on:

- The numbers should be positive integers.
^{(1)} - The difference between the numbers should be even.
^{(2)} - You need to be able to square their average in your head.

My technique leverages your memory of squares to allow you to multiply numbers in your head that are higher than the multiplication tables you had to memorise at school. It’s easier to memorise all the squares up to 30^{2} than the entire multiplication tables up to 30 x 30.

Here’s my multiplication technique in words:

^{2}- (half their difference)

^{2}

Example where x = 13 and y = 17:

(their average)

^{2}= 225

their difference = 4

half their difference = 2

(half their difference)

^{2}= 4

xy = (their average)

^{2}- (half their difference)

^{2}

xy = 225 - 4

xy = 221

13 x 17 = 221

I don’t know about you but I haven’t memorised 13 x 17, but I do know that 15^{2} = 225, and I can halve 4 to 2, square it back to 4, and subtract it from 225 pretty easily in my head. Again, if you know your squares this technique is very useful. If you need a calculator to work out the squares then it’s useless to you.

While writing all this down just now I’ve realised that my technique can be explained using something I learnt in high school algebra called the ‘difference of squares’:

^{2}- b

^{2}

Relating this to my technique and assuming x < y:

(a + b) is y

a is the average of x and y

b is half their difference

My multiplication technique is just a way of taking two numbers and fitting them into the ‘difference of squares’ format. So I didn’t actually discover anything new, I just discovered something new to me that was probably first discovered hundreds of years ago.