A reversible [3n+1 problem] system
In both the ordinary 3n+1 problem and in the systems discussed in the main text different numbers often evolve to the same value so that there is no unique way to reverse the evolution. However, with the rule
n -> If[EvenQ[n], 3n/2, Round[3n/4]]
it is always possible to go backwards by the rule
n -> If[Mod[n,3]==0, 2n/3, Round[4n/3]]
The picture shows the number of base 10 digits in numbers obtained by backward and forward evolution from n=8. For n<8, the system always enters a short cycle. Starting at n=44, there is also a length 12 cycle. But apart from these cycles, the numbers produced always seem to grow without bound at an average rate of 3/(2 Sqrt) in the forward direction, and 2 4^(1/3)/3 in the backward direction (at least all numbers up to 10,000 grow to above 10^100). Approximately one number in 20 has the property that evolution either backward or forward from it never leads to a smaller number.