  What is the maximum possible value of the lead coefficient of a degree $d$
polynomial $Q(x)$ if $|Q(1)|,|Q(2)|,\ldots,|Q(k)|$ are all less than or equal
to one? More generally we write $L_{d,[x_k]}(x)$ for what we prove to be the
unique degree $d$ polynomial with maximum lead coefficient when bounded between
$1$ and $-1$ for $x\in [x_k]=\{x_1,\cdots,x_k\}$. We calculate explicitly the
lead coefficient of $L_{d,[x_k]}(x)$ when $d\leq 4$ and the set $[x_k]$ is an
arithmetic progression. We give an algorithm to generate $L_{d,[x_k]}(x)$ for
all $d$ and $[x_k]$.
