Lawrence J. Thaden
Registered: Jan 2004
Posts: 357 
Emulating Elementary Rule 73 Behavior
Each of the following 243 three color rule numbers exactly reproduces elementary two color rule number 73 behavior.
Every group of three rule numbers is 243 rules distant from the next group of three. Within each group the rule numbers are 9 rule numbers apart.
Adding And[(1 p), (1 q), r] modulo 3 to each rule within a group gives the next rule number.
Adding And[(1 p), (q + 1), r] modulo 3 to the first rule within a group gives the first rule number of the next group.
FoldList[Plus, #, Table[9, {2}]]&/@FoldList[Plus, 3824438629078, Table[243, {80}]];
There is a second and third set of 243 rules with the same distribution as the above set. They start with rule numbers 3824438806225 and 3824438983372 respectively.
There are 157689 rule numbers between the last of the first set and the first of the second set. Again there are 157689 rule numbers between the last of the second set and the first of the third set.
I believe that the second and third sets are additional instances of the three color rule and not directly related to the elementary two color rule.
You can find these three color rules that emulate the elementary two color rules by first finding the three color rule with the same logic expression as the two color rule. For instance elementary two color rule 73 has the same logic expression as three color rule 3824437479971.
However, usually this three color rule will not have a behavior that emulates the elementary rule.
To find the first rule in the set of 243 rules that emulates the elementary behavior you have to search the area immediately before and after the three color rule that has the same logic expression.
So far I have found it is not too distant. And curiously enough it appears to be always a number of rules that is the product of two primes.
For example, with the case at hand we have: 38244386290783824437479971 = (28027 * 41).
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L. J. Thaden
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