  In 1985, Barnsley and Harrington defined a ``Mandelbrot Set'' $\mathcal{M}$
for pairs of similarities --- this is the set of complex numbers $z$ with
$0<|z|<1$ for which the limit set of the semigroup generated by the
similarities $x \mapsto zx$ and $x \mapsto z(x-1)+1$ is connected.
Equivalently, $\mathcal{M}$ is the closure of the set of roots of polynomials
with coefficients in $\lbrace -1,0,1 \rbrace$. Barnsley and Harrington already
noted the (numerically apparent) existence of infinitely many small ``holes''
in $\mathcal{M}$, and conjectured that these holes were genuine. These holes
are very interesting, since they are ``exotic'' components of the space of (2
generator) Schottky semigroups. The existence of at least one hole was
rigorously confirmed by Bandt in 2002, and he conjectured that the interior
points are dense away from the real axis. We introduce the technique of traps
to construct and certify interior points of $\mathcal{M}$, and use them to
prove Bandt's Conjecture. Furthermore, our techniques let us certify the
existence of infinitely many holes in $\mathcal{M}$.
