  We analyze Jim Propp's P-machine, a simple deterministic process that
simulates a random walk on $Z^d$ to within a constant. The proof of the error
bound relies on several estimates in the theory of simple random walks and some
careful summing. We mention three intriguing conjectures concerning
sign-changes and unimodality of functions in the linear span of $\{p(\cdot,x) :
x \in Z^d\}$, where $p(n,x)$ is the probability that a walk beginning from the
origin arrives at $x$ at time $n$.
