$Z$-Cyclic $Wh(28)$

Abstract

Whist tournament designs are known to exist for all $v\equiv 0,1(\mathrm{mod}4)$. Much less is known about the existence of $\mathbb{Z}$-cyclic whist designs. Previous studies $[5,6]$ have reported on all $\mathbb{Z}$-cyclic whist designs for $v\in \{4,5,8,9,12,13,16,17,20,21,24,25\}$. This paper is a report on all $\mathbb{Z}$-cyclic whist tournament designs on 28 players, including a detailed summary of all known whist specializations related to a 28 player $\mathbb{Z}$-cyclic whist design. Our study shows that there are $7,910,127$ $\mathbb{Z}$-cyclic whist designs on 28 players. Of these designs, $2,568,510$ possess the Three Person Property, $240,948$ possess the Triplewhist Property and none possess the Balancedwhist Property. Introduced here is the concept of the mirror image of a $\mathbb{Z}$-cyclic whist design. In general, utilization of this concept reduces the computer search for $\mathbb{Z}$-cyclic whist designs by nearly fifty percent.