  We study two-player \emph{take-away} games whose outcomes emulate two-state
one-dimensional cellular automata, such as Wolfram's rules 60 and 110. Given an
initial string consisting of a central data pattern and periodic left and right
patterns, the rule 110 cellular automaton was recently proved Turing-complete
by Matthew Cook. Hence, many questions regarding its behavior are
algorithmically undecidable. We show that similar questions are undecidable for
our \emph{rule 110} game.
