This problem is from Niven's, 5.4.8.
Let $\,f(\mathbf{x})=f(x_1,x_2,x_3) = x_1^4+x_2^4+x_3^4-x_1^2x_2^2-x_2^2x_3^2-x_3^2x_1^2-x_1x_2x_3(x_1+x_2+x_3).$ Show that $f(\mathbf{x}) \equiv1$ $($mod$\,4$$)$ unless all three variables are even. Deduce that if $$\, f(\mathbf{x})+f(\mathbf{y})+f(\mathbf{z}) = 4(f(\mathbf{u})+f(\mathbf{v})+f(\mathbf{w}))$$ for integral values of the variables, then all $18$ variables are $0$.
Of course, one way to do that is to consider the remainders modulo $4$ of an integer. But that would be horrible, so I thought maybe I could group them in a nice way and then consider the remainders modulo $4$. I did try hard, but not even close :( so I really need some hints.