Question 145
Let $T$ be the set of all triples $(p, q, r)$ of nonnegative integers. Determine all functions $f: T \rightarrow \mathbb{R}$ for which applies
$$
f(p, q, r)= \begin{cases}0 & \text{for} p q r=0 \\ 1+\frac{1}{6}\{f(p+1, q-1, r)+f(p-1, q+1, r) \\ +f(p-1, q, r+1)+f(p+1, q, r-1) \\ +f(p, q+1, r-1)+f(p, q-1, r+1)\} & \text{otherwise.}\end{cases}
$$
$$
f(p, q, r)= \begin{cases}0 & \text{for} p q r=0 \\ 1+\frac{1}{6}\{f(p+1, q-1, r)+f(p-1, q+1, r) \\ +f(p-1, q, r+1)+f(p+1, q, r-1) \\ +f(p, q+1, r-1)+f(p, q-1, r+1)\} & \text{otherwise.}\end{cases}
$$