  Given a finite nonempty sequence S of integers, write it as XY^k, where Y^k
is a power of greatest exponent that is a suffix of S: this k is the curling
number of S. The Curling Number Conjecture is that if one starts with any
initial sequence S, and extends it by repeatedly appending the curling number
of the current sequence, the sequence will eventually reach 1. The conjecture
remains open. In this paper we discuss the special case when S consists just of
2's and 3's. Even this case remains open, but we determine how far a sequence
of n 2's and 3's can extend before reaching a 1, conjecturally for n <= 80. We
investigate several related combinatorial problems, such as finding c(n,k), the
number of binary sequences of length n and curling number k, and t(n,i), the
number of sequences of length n which extend for i steps before reaching a 1. A
number of interesting combinatorial problems remain unsolved.
