  We are investigating who has the winning strategy in a game in which two
players take turns drawing arrows trying to complete cycle cells in a graph. A
cycle cell is a cycle with no chords. We examine game boards where the winning
strategy was previously unknown. Starting with a $C_{5}$ sharing two
consecutive edges with a $C_{7}$ we solve multiple classes of graphs involving
"stacked" polygons. We then expand upon and improve previous theorems and
conjectures, and offer some new directions of research related to the Game of
Cycles. The original game was described by Francis Su in his book Mathematics
for Human Flourishing. The first results on the game were published in The Game
of Cycles arXiv:arch-ive/04.00776.

Combinatorics