Searching [for doubling rules]

No symmetric k = 3, r = 1 rule yields doubling. General rules can show subtle bugs; rule 1340716537107 for example first fails at n = 24. The total number of k = 3, r = 1 rules that need to be searched can easily be reduced from 3^{27} to 3^{21}. Several different rules that work can behave identically, since up to 6 of the 27 cases in each rule are not sampled with the initial conditions used. In rules that work, between 8 and 19 cases lead to a change in the color of a cell, with 14 cases being the most common.