This question is hard but doable. two sides of a (non-degenerate) triangle are 11 and 17. how many possible lengths are there for the third side, if it is a positive integer?

By the triangle inequality, the smaller two sides' lengths must add up to a number larger than the length of the largest side. So either



or



In other words, either is not the largest side length, or it is.

Taken together, we must have , which yields 22 integer solutions.

17 - 11 = 6
17 + 11 = 28
6 < third side <28

so any side from 7 to 27 are possible lengths

answer
there are 21 possible lengths for third side if it is a positive integer