  In this paper we present two independent computational proofs that the monoid
derived from $5\times 5\times 3$ contingency tables is normal, completing the
classification by Hibi and Ohsugi. We show that Vlach's vector disproving
normality for the monoid derived from $6\times 4\times 3$ contingency tables is
the unique minimal such vector up to symmetry. Finally, we compute the full
Hilbert basis of the cone associated with the non-normal monoid of the
semi-graphoid for $|N|=5$. The computations are based on extensions of the
packages LattE-4ti2 and Normaliz.
