Question 1
Let $T$ denote the set of all ordered triples $(p, q, r)$ of nonnegative integers. Find all functions $f: T \to \mathbb{R}$ such that
$$f(p,q,r)=\begin{cases}0,&pqr=0,\\1+\frac16\bigl(f(p+1,q-1,r)+f(p-1,q+1,r)+f(p-1,q,r+1)+f(p+1,q,r-1)+f(p,q+1,r-1)+f(p,q-1,r+1)\bigr),&pqr\ne0.\end{cases}$$
$$f(p,q,r)=\begin{cases}0,&pqr=0,\\1+\frac16\bigl(f(p+1,q-1,r)+f(p-1,q+1,r)+f(p-1,q,r+1)+f(p+1,q,r-1)+f(p,q+1,r-1)+f(p,q-1,r+1)\bigr),&pqr\ne0.\end{cases}$$