Impartial games emulating one-dimensional cellular automata and undecidability

We study two-player impartial games whose outcomes emulate two-state one-dimensional cellular automata, such as Wolframʼs 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 rule 110 games.