Question 540914
Lucky for us, we got those 3 numbers.
The problem tells us that for 370, the sum of the cubes of the digits is
{{{3^3+7^3+0^3=3^3+7^3=370}}}
Then, for 371, the sum of the cubes of the digits must be
{{{3^3+7^3+1^3=3^3+7^3+1=371}}}
BINGO!
The fourth number is 371.
No, I did not realize immediately.
I found it through a lot of logical work
After I found it, I immediately realized that it was so obvious (in hindsight) that if one of the solutions ended in zero, the next integer would also be a solution.
That is the kind of question that would be easily answered by a 13-year old in the artofproblemsolving forum. (That's full of kids with at least semi-realistic hopes to get to the Math Olympiad).