  As a strategy to complete games quickly, we investigate one-dimensional
random walks where the step length increases deterministically upon each return
to the origin. When the step length after the kth return equals k, the
displacement of the walk x grows linearly in time. Asymptotically, the
probability distribution of displacements is a purely exponentially decaying
function of |x|/t. The probability E(t,L) for the walk to escape a bounded
domain of size L at time t decays algebraically in the long time limit, E(t,L)
~ L/t^2. Consequently, the mean escape time <t> ~ L ln L, while <t^n> ~
L^{2n-1} for n>1. Corresponding results are derived when the step length after
the kth return scales as k^alpha$ for alpha>0.
