Question 3
For a positive integer $n$, put $S_n=\{0,1,\ldots,2n+1\}$. Let $F$ be the set of all functions $f:\mathbb Z\times S_n\to[0,1]$ satisfying
(i) $f(x,0)=f(x,2n+1)=0$ for every integer $x$;
(ii) $f(x-1,y)+f(x+1,y)+f(x,y-1)+f(x,y+1)=1$ for integers $x$ and $1\le y\le2n$.
1. Prove that $F$ is infinite.
2. For $f\in F$, let $v_f$ be the image of $f$. Prove that $v_f$ is finite.
3. Find the largest possible value of $|v_f|$.
(i) $f(x,0)=f(x,2n+1)=0$ for every integer $x$;
(ii) $f(x-1,y)+f(x+1,y)+f(x,y-1)+f(x,y+1)=1$ for integers $x$ and $1\le y\le2n$.
1. Prove that $F$ is infinite.
2. For $f\in F$, let $v_f$ be the image of $f$. Prove that $v_f$ is finite.
3. Find the largest possible value of $|v_f|$.