Conditional number problem

I am assisting a programming comp, and our team had a practice question for finding values a,b,c such that:

a^3 - b^3 - c^3 = 100

Currently, we had the idea of doing random number tests which squares, by restricting the possibilities for the numbers based on size and nature (obviously we had to end with an even number, etc). Additionally, we tried doing random testing for a (i.e subbing a value in) and then making the condition b^3 + c^3 = a^3 - 100 to find appropriate b and c values through more testing. However, both methods, as you image, take too long. We have a limit of about 10 seconds so some guessing is possible.

I am really stumped and would appreciate any help.