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Cube Roots Whizz

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.

Trick Explanations


Martin Gardner,
Mathematical Carnival,
Peguin Books 1990