For the vast majority of rules written down at random, such problems do indeed occur. But it is possible to find rules in which they do not, and the pictures on the previous two pages [129, 130] show a few examples I have found of such rules. In cases (a) and (b), the behavior is fairly simple. But in the other cases, it is considerably more complicated.

There is a steady overall increase, but superimposed on this increase are fluctuations, as shown in the pictures on the facing page.

In cases (c) and (d), these fluctuations turn out to have a very regular nested form. But in the other cases, the fluctuations seem instead in many respects random. Thus in case (f), for example, the number of positive and negative fluctuations appears on average to be equal even after a million steps.

But in a sense one of the most surprising features of the facing page is that the fluctuations it shows are so violent. One might have thought that in going say from f[2000] to f[2001] there would only ever be a small change. After all, between n=2000 and 2001 there is only a 0.05% change in the size of n.

But much as we saw in the previous section it turns out that it is not so much the size of n that seems to matter as various aspects of its representation. And indeed, in cases (c) and (d), for example, it so happens that there is a direct relationship between the fluctuations in f[n] and the base 2 digit sequence of n.

In case (d), the fluctuation in each f[n] turns out to be essentially just the number of 1's that occur in the base 2 digit sequence for n. And in case (c), the fluctuations are determined by the total number of 1's that occur in the digit sequences of all numbers less than n.

There are no such simple relationships for the other rules shown on the facing page. But in general one suspects that all these rules can be thought of as being like simple computer programs that take some representation of n as their input.

And what we have discovered in this section is that even though the rules ultimately involve only addition and subtraction, they nevertheless correspond to programs that are capable of producing behavior of great complexity.