## Friday, 22 May 2015

### Calculating cube roots in your head

I thought I would swing over to mathematics for a bit.

You don't need a real aptitude to enjoy maths, and even if you had woefully useless teachers (as I did),  you can still have fun with numbers.

Are you one of those people who play with numbers? I was, even was I was small, and when I was supposed to be sleeping, I would be lying in bed, going "2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 . . ." on and on, seeing how high I could go.

I usually got 22 or 23 terms before I lost track and had to start again. I just enjoy looking for patterns (or I fell asleep).

Sometimes, I find something useful, like a way to fool people into thinking that I had memorised the cubes of all the numbers from 100 to 200.

I developed this trick by looking at the final digit of various cubes:

If you cube a number ending in 1 the result ends in 1
If you cube a number ending in 2 the result ends in 8
If you cube a number ending in 3 the result ends in 7
If you cube a number ending in 4 the result ends in 4
If you cube a number ending in 5 the result ends in 5
If you cube a number ending in 6 the result ends in 6
If you cube a number ending in 7 the result ends in 3
If you cube a number ending in 8 the result ends in 2
If you cube a number ending in 9 the result ends in 9
and obviously,
If you cube a number ending in 0 the result ends in 0

See the pattern? If you know the last digit of the cube, you know the last digit of the seed number that was cubed.

Then I realised that you can memorise the approximate ranges for the cubes of 100 to 109, 110 to 119 and so on:
100 – 109 1 to 1.3 million
110 – 119 1.3 to 1.7 million
120 – 129 1.7 to 2.2 million
130 – 139 2.2 to 2.7 million
140 – 149 2.7 to 3.3 million
150 – 159 3.4 to 4 million
160 – 169 4.1 to 4.8 million
170 – 179 4.9 to 5.7 million
180 – 189 5.8 to 6.7 million
190 – 199 6.8 to 8 million

so if I hear 1442897, I know from the first two digits that we are between 110 and 119, and the last digit of the cube (7) tells me the last digit of the number being cubed is 3, so the answer is 113.

Try it!