What is a number that is equal to the sum of the digits of its cube?

Answer 1

Let's start with the case where it's one digit. In this case, all we have to do is solve for the equation
x^3 = x, and we get
x^3 - x = 0
x(x^2 - 1) = 0
x(x - 1)(x + 1) = 0
x = {1, -1, 0}

We discard x = -1 because we can't have -1 digits. We also can't have 0 because we can't have 0 digits in a number. Therefore, our answer is 1.

Answer 2

1 or 0

Answer 3

0 and 1

Answer 4

You want numbers other than the trivial 0 and 1, I am sure...

There are four of them: 153, 370, 371, and I forget the last one.

* * * * *

OK, the last one is 407. These are known as the three-digit "narcissistic numbers" - the sum of the digits cubed equals the number. You can extend this to four-digit numbers, i.e. the sum of the digits raised to the 4th power equals the number; there are five naricissistic 4-digit numbers. The largest narcissistic number is 39 digits long - the sum of the digits raised to the 39th power equals the number.

Yes, I know that was w-a-y more than you cared to know about narcissistic numbers. :)

Answer 5

one!