Summary: Let \(p=2^kt+1\) be a prime where \(t>1\) is an odd integer, \(k\geq 2\). Methods of constructing a \(Z\)-cyclic triple whist tournament \(\text{TWh}(p)\) are given. By such methods we construct a \(Z\)-cyclic \(\text{TWh}(p)\) for all primes \(p\), \(p\equiv 1\pmod 4\), \(29\leq p\leq 16097\), except \(p=257\). Let \(p_i=2^{k_i} t_i +1\), \(q=2^{k_0}t_0+3\) be primes where \(t_i\); \(i=0,1,\dots,n\) are odd \(>1\) and \(k_i\) are integers \(\geq 2\). We prove that if \(Z\)-cyclic \(\text{TWh}(p_i)\) and \(\text{TWh}(q+1)\) exist then \(Z\)-cyclic \(\text{TWh}(\prod^n_{i=1}p_i^{\alpha_i})\) and \(\text{TWh} (q\prod^n_{i=1}p_i^{\alpha_i}+1)\) exist.