VN-TST 2013 Problem 3

Given a number $n\in\mathbb{Z}^+$ and let $S$ denotes the set $\{0,1,2,...,2n+1\}$ . Consider the function $f:\mathbb{Z}\times S\to [0,1]$ satisfying two following conditions simultaneously:

i) $f(x,0)=f(x,2n+1)=0\forall x\in\mathbb{Z}$ ; ii) $f(x-1,y)+f(x+1,y)+f(x,y-1)+f(x,y+1)=1$ for all $x\in\mathbb{Z}$ and $y\in\{1,2,3,...,2n\}$ .

Let $F$ be the set of such functions. For each $f\in F$ , let $v(f)$ be the set of values of $f$ .

a) Proof that $|F|=\infty$ .

b) Proof that for each $f\in F$ then $|v(f)|<\infty$ .

c) Find the maximum value of $|v(f)|$ for $f\in F$ .