Question 15
Find the number of functions $f$ from $\{0, 1, 2, 3, 4, 5, 6\}$ to the integers such that $f(0) = 0,$ $f(6) = 12,$ and $$
\begin{aligned} |x - y| &\le |f(x) - f(y)| \\ &\le 3|x - y| \end{aligned}
$$ for all $x$ and $y$ in $\{0, 1, 2, 3, 4, 5, 6\}.$
\begin{aligned} |x - y| &\le |f(x) - f(y)| \\ &\le 3|x - y| \end{aligned}
$$ for all $x$ and $y$ in $\{0, 1, 2, 3, 4, 5, 6\}.$