It depends on what did you mean by saying perfect square. If I've understood it correctly, I can help you with a part of your problem. The squares of mod 9 are 1,4,7 which are came from 1,2,4. Addition of the given numbers are 2,3,5,6, 8, which are exactly the part of your problem. This number, which is not shown as squares Mod 9, and thus doesn't appear as a sum of digits of a perfect square. I hope you will find it helpful.
