A question from Number Theory book of Thomas Koshy?

I have a question here from the book I have mentioned above

The question is - There are 4 nos between 100 and 1000, which are equal to the sum of the cubes of its digits. Three of them are 153, 371 and 407. Find the fourth no.

The answer is 370.

I still actually didn't get as to how to proceed to solve the problem. Can anyone explain the solution to me.

I found a pattern here - In each of the nos, the transition of the nos from one to the next, results in the change of only one digit

153

} 7

371

} 1

370

} 4

407

But I need a more plausible explanation as to how I can go about solving this and not use mere brute force!