  The fundamental group of the Menger universal curve is uncountable and not
free, although all of its finitely generated subgroups are free. It contains an
isomorphic copy of the fundamental group of every one-dimensional separable
metric space and an isomorphic copy of the fundamental group of every planar
Peano continuum. We give an explicit and systematic combinatorial description
of the fundamental group of the Menger universal curve and its generalized
Cayley graph in terms of word sequences. The word calculus, which requires only
two letters and their inverses, is based on Pasynkov's partial topological
product representation and can be expressed in terms of a variation on the
classical puzzle known as the Towers of Hanoi.

Geometric Topology