  We study one-dimensional cellular automata whose rules are chosen at random
from among $r$-neighbor rules with a large number $n$ of states. Our main focus
is the asymptotic behavior, as $n \to \infty$, of the longest temporal period
$X_{\sigma,n}$ of a periodic solution with a given spatial period $\sigma$. We
prove, when $\sigma \le r$, that this random variable is of order
$n^{\sigma/2}$, in that $X_{\sigma,n}/n^{\sigma/2}$ converges to a nontrivial
distribution. For the case $\sigma > r$, we present empirical evidence in
support of the conjecture that the same result holds.
