Friday, March 14, 2003 - 06:08 PM

If someone cubed a two-digit number on a calculator and gave you the result - but not the original number - could you extract the cube root? With this trick, you'll be able to do just that - instantly!

A bit of homework is required for this trick, but it's worth the effort if you like to show off.

First, memorize the cubes of the digits 1 through 9:

1, 8, 27, 64, 125, 216, 343, 512, 729.

Next memorize the "endings" of the cubes. For example, the ending of 9³ is 9, because 9³ = 729. The "ending" (or last digit) is 9.

So let's make a list. "1 cubed ends in 1" is abbreviated "1 --> 1".

1 --> 1
2 --> 8
3 --> 7
4 --> 4
5 --> 5
6 --> 6
7 --> 3
8 --> 2
9 --> 9

These are easily memorized. 1 and 9 (at the extremes) are "self-enders", as are the 4, 5, and 6 (in the center). The others involve "a sum of 10": 2 ends in 8, 8 ends in 2, 3 ends in 7, and 7 ends in 3.

Now how to do the trick!
Tell a friend to secretly pick any two-digit number and then have him or her use a calculator to cube it. Let's say he picks 76. So using the calculator he computes 76 x 76 x 76 . He then tells you the cube: 438,976.

To instantly determine his original number (ie, compute the cube root), follow these easy steps:

1. Drop the last three digits and find the largest cube contained in 438. This is 7³ = 343, so the tens-digit is 7.
   (This is why you had to memorize the cubes of the digits 1 through 9)
2. Now go back to the last three digits. Look at the last digit, 6. That's the same ending as 6³, so your units-digit is 6.
   (This is why you had to memorize the "endings" of the cubes for digits 1 through 9)

So the cube root of 438,976 is 76

Another example:
Let's say your friend chooses a secret two-digit number whose cube is 79,507. How do you instantly determine the cube root?

1. Drop the last three digits and find the largest cube in 79. This is 4³ = 64, so the tens-digit of the cube root is 4.
2. Now go back to the last three digits. Look at the last digit, 7. That's the same ending as 3-cubed. So the units-digit of your cube root is 3.

Therefore, the cube root of 79,507 is 43.

A very special thanks to Bob Kurosaka, retired math professor, for generously sharing this trick!

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