  In his dissertation, Wadge defined a notion of guessability on subsets of the
Baire space and gave two characterizations of guessable sets. A set is
guessable iff it is in the second ambiguous class (boldface Delta^0_2), iff it
is eventually annihilated by a certain remainder. We simplify this remainder
and give a new proof of the latter equivalence. We then introduce a notion of
guessing with an ordinal limit on how often one can change one's mind. We show
that for every ordinal alpha, a guessable set is annihilated by alpha
applications of the simplified remainder if and only if it is guessable with
fewer than alpha mind changes. We use guessability with fewer than alpha mind
changes to give a semi-characterization of the Hausdorff difference hierarchy,
and indicate how Wadge's notion of guessability can be generalized to
higher-order guessability, providing characterizations for boldface
Delta^0_alpha for all successor ordinals alpha>1.
