  A compact packing is a set of non-overlapping discs where all the holes
between discs are curvilinear triangles. There is only one compact packing by
discs of size $1$. There are exactly $9$ values of $r$ which allow a compact
packing by discs of sizes $1$ and $r$. We prove here that there are exactly
$164$ pairs $(r,s)$ allowing a compact packing by discs of sizes $1$, $r$ and
$s$.
