If we take any two-digit number, and multiply it by itself and then multiply the result by the first number again, we are said to have found the CUBE of this number. For example, 63 x 63 x 63 = 250,047 which is 63 cubed.
This part of the calculation is boring and you will probably need a calculator. However, believe it or not, it is possible for someone to hear the result of this calculation and immediately tell you which number you started with! It takes a little bit of practice, but in fact the method for doing this is fairly simple.
How is this done?
First, we need to take the cubes of all of the numbers from 1 to 9:
1 x 1 x 1 = 1
2 x 2 x 2 = 8
3 x 3 x 3 = 27
4 x 4 x 4 = 64
5 x 5 x 5 = 125
6 x 6 x 6 = 216
7 x 7 x 7 = 343
8 x 8 x 8 = 512
9 x 9 x 9 = 729
Now, imagine someone tells you the number they have found by cubing a two digit number. This number will be some amount of thousands, and then another part which is some number of hundreds, tens and ones.
The first digit of their original number is given by the number of thousands, by comparing it to the answers in the table above - so, if the number of thousands is between 1 and 7, their number started with a 1. If it is between 8 and 26, their number started with a 2. If the number of thousands is between 27 and 63, it was a three, and so on. You can find which gap in the table the number of thousands fits into, and say it is between the numbers in rows 5 and 6, which are 125 and 216, take the lower of the two, so their first digit was a 5.
The second digit is found by looking at the final digit in the table above. The final digits down the right hand side are in the following order: 1, 8, 7, 4, 5, 6, 3, 2, 9. Here, 8 is swapped with 2 and 7 is swapped with 3, and otherwise the digits pair up with the number of the row. So, if the long number your friend told you ends in a 4, then the second digit they started with was a 4. If it ends in a 3, their starting number ended in a 7 and so on.
For example, above we had 250,047. First we note that 250 lies between 216 and 342, so the first digit of the orignal number must have been a 6. The final digit of 250,047 is 7, which means the second digit of the original number must have been a 3. Hence the number we cubed was 63!
If you memorise these nine numbers, and what their last digits are, with a little bit of practise you can take any cube of a two digit number and find out what the original number was, in a quite short time! Why not try cubing a few numbers and seeing if you can work backwards to get the same number you started with? Then try it on your friends!
Tricks like this are called speed calculations. They can be very impressive, and the reason it works is because the numbers follow a pattern - the last digits follow a pattern, and the first part of the number will always lie within the correct range, provided your original number was two digits. If we can work out what the pattern is, we can easily work backwards