Three cubes

In the picture above, when we calculate the sum of the three cubes on the left side, we see that two of the digits are present in the sum.

There are no 2-digit numbers which are equal to the sum of the squares of their digits.
There are 3-digit numbers which are equal to the sum of their third powers. Which are they?

(My sincerest apologies for not providing a link to a solution. WordPress has changed the interface and I have been unable to master it. I’m still looking for the right way to switch to html editing and add images and links)