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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.

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