I looked a little at the 3N+5 and 3N+7 problems. My impression, based on some hand calculations, is that 3N+5 has more loops than 3N+-1, and 3N+7 has even more. [Only A = +-1(mod 6) are interesting as 3N+A problems.] If we use a prime divisor, branching for example on N (mod 5), then there's an obvious procedure for "deciding" if the overall average drift of the stepping function tends to increase or decrease the numbers, leading to either everything heading toward small numbers, or almost everything diverging. The procedure is to examine the output multipliers. The Collatz problem can be written as 2N -> N 2N+1 -> 6N+4 Any multipliers that are divisible by the input's dispatch prime (2 for Collatz) must have the rule applied to the output: 6N+4 -> 3N+2. (This can lead to infinite loops in some cases, or to fixed numbers like ->5.) In the ordinary case, we wind up with all the multipliers being indivisible by the dispatch prime. Collatz becomes 2N -> N 2N+1 --> 3N+2 Now we abandon rigor and guess that inputs are uniformly distributed mod P (the dispatch prime), and so are outputs. To measure average growth or shrinkage, just multiply together all the output multipliers, and compare with the product of the input multipliers, P^P. For Collatz, this is 2^2 = 4 on the input side, and 1*3 = 3 on the output side. So we expect Collatz to, on average, shrink the input, matching observations. If we do (odd N)->3N+5 instead of 3N+1, it doesn't matter; we still get shrinkage. But 3N+2 is rather different; the preprocessing step diverges. And 5N+1, instead of 3N+1, also diverges on average. (Again matching observation; unfortunately, very little of what I'm saying can be proven.) For composite dispatch moduli such as 4, 6, or 30, things are more complicated. If the output multipliers all happen to be relatively prime to the dispatch modulus, then the same product business applies. But in the more interesting case, some of the output multipliers share a common factor with the input modulus. If the preprocessing step can be completed to eliminate the common factors, all well and good. But sometimes it can't, and this analysis doesn't work. Two-Counter Machines are the natural setting for Collatz puzzles. I learned about CMs from Minsky's automata theory course that evolved into his book Finite & Infinite Machines. I don't know if he gets credit for inventing Counter Machines, but that's where I first encountered them. There's some small multiple of 30, perhaps 1500, which can be used as a dispatch modulus and will simulate a universal turing machine (via a two-counter machine implementation). I've played around some with dispatch moduli 4, 6, and 8, but have no results other than that there's a lot of variety in behavior. It might be possible to come up with a better preprocessing step that works with prime power dispatchers like 4 and 8, but I think 6 needs a new idea. Rich