Let
![T](/media/m/0/1/6/016d42c58f7f5f06bdf8af6b85141914.png)
denote the set of all ordered triples
![\left(p,q,r\right)](/media/m/a/8/2/a82f10a657e8c40f9ba253e70b61dede.png)
of nonnegative integers. Find all functions
![f: T \rightarrow \mathbb{R}](/media/m/7/0/f/70ff5f6321b72c7bd018fe0059976168.png)
satisfying
for all nonnegative integers
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
,
![q](/media/m/c/1/d/c1db9b1124cc69b01f9a33595637de69.png)
,
![r](/media/m/3/d/f/3df7cc5bbfb7b3948b16db0d40571068.png)
.
%V0
Let $T$ denote the set of all ordered triples $\left(p,q,r\right)$ of nonnegative integers. Find all functions $f: T \rightarrow \mathbb{R}$ satisfying
$$f(p,q,r) =
\begin{cases}
0 &\text{if}\; pqr = 0,\\
1+\frac{1}{6}(f(p+1,q-1,r)+f(p-1,q+1,r) &\\
+f(p-1,q,r+1)+f(p+1,q,r-1) &\\
+f(p,q+1,r-1)+f(p,q-1,r+1)) &\text{otherwise}\end{cases}$$
for all nonnegative integers $p$, $q$, $r$.