8. $[\mathbf{5}]$ Determine the largest positive integer $n$ such that there exist positive integers $x, y, z$ so that
$$
n^{2}=x^{2}+y^{2}+z^{2}+2 x y+2 y z+2 z x+3 x+3 y+3 z-6
$$
Answer: 8. The given equation rewrites as $n^{2}=(x+y+z+1)^{2}+(x+y+z+1)-8$. Writing $r=x+y+z+1$, we have $n^{2}=r^{2}+r-8$. Clearly, one possibility is $n=r=8$, which is realized by $x=y=1, z=6$. On the other hand, for $r>8$, we have $r^{2}<r^{2}+r-8<(r+1)^{2}$.