Question 39
For each positive integer $n$, let $S_n = \{0,1,2,\ldots,2n+1\}$. Consider the function $f : (\mathbb{Z} \times S_n) \to [0;1]$ satisfying the following conditions:
i/ $f(x,0) = f(x,2n+1) = 0$ for all $x$ from integers.
ii/ $f(x-1,y)+f(x+1,y)+f(x,y-1)+f(x,y+1)=1$ for all $x,y \in \mathbb{Z}$ and $1 \le y \le 2n$.
Let $F$ be the set of all such functions.
1. Prove that $|F|$ is infinite.
2. For each function $f \in F$, let $v_f$ be the image set of $f$. Prove that $|v_f|$ is finite.
3. Find the maximum value of $|v_f|$ where $f \in F$.
i/ $f(x,0) = f(x,2n+1) = 0$ for all $x$ from integers.
ii/ $f(x-1,y)+f(x+1,y)+f(x,y-1)+f(x,y+1)=1$ for all $x,y \in \mathbb{Z}$ and $1 \le y \le 2n$.
Let $F$ be the set of all such functions.
1. Prove that $|F|$ is infinite.
2. For each function $f \in F$, let $v_f$ be the image set of $f$. Prove that $|v_f|$ is finite.
3. Find the maximum value of $|v_f|$ where $f \in F$.