For clarity, let's summarize why mod 4 doesn't work. We know any square is 0 or 1 (mod 4). Thus the sum of 3 squares is either 0, 1, 2, or 3 (mod 4). But since this is all possibilities it is not very helpful.

For purposes of this problem, we could say 4 is not "large enough" as a mod. Thinking about your answers from 5.26 (where you listed possible squares mod m for m < 10) see if you could find another mod that might work.

Hope this helps a bit!