  We develop an exactly-solvable three-state discrete-time minimal Brownian
ratchet (MBR), where the transition probabilities between states are
asymmetric. By solving the master equations we obtain the steady-state
probabilities. Generally the steady-state solution does not display detailed
balance, giving rise to an induced directional motion in the MBR. For a reduced
two-dimensional parameter space we find the null-curve on which the net current
vanishes and detailed balance holds. A system on this curve is said to be
balanced. On the null-curve, an additional source of external random noise is
introduced to show that a directional motion can be induced under the zero
overall driving force. We also indicate the off-balance behavior with biased
random noise.
