  What is the maximum number of holes that a polyomino with $n$ tiles can
enclose? Call this number $f(n)$. We show that if $n_k = \left( 2^{2k+1} + 3
\cdot 2^{k+1}+4 \right) / 3$ and $h_k = \left( 2^{2k}-1 \right) /3$, then
$f(n_k) = h_k$ for $k \ge 1$. We also give nearly matching upper and lower
bounds for large $n$, showing as a corollary that $f(n) \approx n/2$.
