Rich Schroeppel writes:
Have you looked at whether varying the split ratio also gives the odd lower bound results? What about 1:3 split (with various rounding rules), or 1:4 split, or 1:2:2; or 1:1:1, or fiddling with the reach 1:*:1:1?
Nice idea. As you noticed, the 1:3 split requires a tie-breaking rule for rounding numbers halfway between two integers, so I'll skip over that (since I don't know what the right tie-breaking rule should be --- though I have a strong conviction that the sweetest general theory will be one in which number of particles is conserved over time). The 1:4 split gives the sequence 1, 2, 2, 3, 4, 5, 6, 6, 7, ... and I notice right off the bat that, although the asymptotic value of the nth term is 3n/4, some of the terms fall short of this value; for instance, the 3rd term is less than 9/4. There may still be an asymmetry here, but it's not as dramatic as in the 1:2 split case. I'm not sure what you mean by 1:2:2. I'm guessing the idea is that nearly 1/5 go left, nearly 2/5 go right, and nearly 2/5 stay put, but I don't know how to do the rounding, since the integers closest to n/5, 2n/5, and 2n/5 don't always add up to n (e.g., take n=1), and I'm extremely prejudiced against non-conservative rules. Jim