  For a finitely presented group, the word problem asks for an algorithm which
declares whether or not words on the generators represent the identity. The
Dehn function is a complexity measure of a direct attack on the word problem by
applying the defining relations. Dison & Riley showed that a "hydra phenomenon"
gives rise to novel groups with extremely fast growing (Ackermannian) Dehn
functions. Here we show that nevertheless, there are efficient (polynomial
time) solutions to the word problems of these groups. Our main innovation is a
means of computing efficiently with enormous integers which are represented in
compressed forms by strings of Ackermann functions.
