I'm a little surprised no one has pointed out yet that this is incredibly close to cellular automata, which Wolfram has show can get you Turing completeness among other things.[1]

Wolfram's "New Kind of Science" almost certainly has tons of fun things for the author to play around with if so inclined.

[1] https://en.m.wikipedia.org/wiki/A_New_Kind_of_Science#Simple...