  By encoding configurations of the ultra-discrete Toda lattice by piecewise
linear paths whose gradient alternates between $-1$ and $1$, we show that the
dynamics of the system can be described in terms of a shifted version of
Pitman's transformation (that is, reflection in the past maximum of the path
encoding). This characterisation of the dynamics applies to finite
configurations in both the non-periodic and periodic cases, and also admits an
extension to infinite configurations. The latter point is important in the
study of invariant measures for the ultra-discrete Toda lattice, which is
pursued in a parallel work. We also describe a generalisation of the result to
a continuous version of the box-ball system, whose states are described by
continuous functions whose gradient may take values other than $\pm1$.
