Start with the set $\{3, 4, 12\}$. In each step you may choose two of the numbers $a$, $b$ and replace them by $0.6a − 0.8b$ and $0.8a + 0.6b$. Can you reach $\{4, 6, 12\}$ in finitely many steps:

Invariant here is that $a^2+b^2$ remains constant. Till here I am good This part of the solution i didn't understand. Since $a^2 +b^2 +c^2= 3^2 +4^2 +12^2 =13^2$ , the point $(a, b, c)$ lies on the sphere around $O$ with radius $13$. Because $4^2 +6^2 +12^2= 14^2$ , the goal lies on the sphere around $O$ with radius $14$. The goal cannot be reached. please explain Will it make any difference if z-points are different