Groups and the Group Axioms · Hard problem

Consider every way to fill in a $4\times4$ table with entries from $\{0,1,2,3\}$ so that row $0$ and column $0$ read $0,1,2,3$ in order, and every row and every column still contains each of $0,1,2,3$ exactly once (the Latin square property). There are exactly $4$ such tables. Of these $4$ candidate tables, how many are ALSO associative, i.e. could actually be the Cayley table of a group?