  We construct a stationary ergodic process $X_1, X_2, \ldots $ such that each
$X_t$ has the uniform distribution on the unit square and the length $L_n$ of
the shortest path through the points $X_1, X_2, \ldots,X_n$ is not asymptotic
to a constant times the square root of $n$. In other words, we show that the
Beardwood, Halton and Hammersley theorem does not extend from the case of
independent uniformly distributed random variables to the case of stationary
ergodic sequences with uniform marginal distributions.
