Summary: Recent results due to Anderson, Finizio and Leonard simplify, considerably, the problem of obtaining Z-cyclic whist tournaments in general and Z-cylic triplewhist tournaments in particular. Their results are of a product nature in that they build composite Z-cyclic tournaments from component Z-cyclic tournaments. It is a fact that Z-cyclic triplewhist tournaments do not exist when the number of players is either 5, 13 or 17. Thus if the total number of players in a Z-cyclic triplewhist tournament is divisible by any of 5, 13, 17 then the Anderson et al. results do not, in general, apply. In this study we focus on two classes of problems that are not obtainable via the results of Anderson et al. Specifically we obtain Z-cyclic triplewhist tournaments on \(v\) players when \(v\) is of one of the forms \(3qp_1p_2\) or \(3qp_1p_2p_3\) with \(q\in \{7,11\}\) and \(p_i\in \{5,13,17\}\). These results coupled with those of Anderson et al. lead to an existence result for Z-cyclic triplewhist tournaments on \(3qp^{\alpha_1}_1\cdots p^{\alpha_n}_n\) players, \(p_i\) prime, \(p_i\equiv 1\pmod 4\), \(q\in \{7,11\}\).