See The smallest sum.

1. Show that there are no **positive** integers {X,Y,Z} (with Z less than Y less than X) such that X+Y, X-Y, X+Z, X-Z, Y+Z, Y-Z are all squares; or provide a counter-example.

2. Assuming that no counter-example exists, what is the minimum such set {X,Y,Z} for which each of X+Y, X-Y, X+Z, X-Z, and either of Y+Z or Y-Z, are all squares?