We review Alekseyev’s method for obtaining the mean of the hitting time of an n-disk randomly moving Hanoi Tower (RMHT) for certain initial and final conditions.  We also give a new method for finding the mean time it takes a RMHT to transfer all n of its disks from peg 1 to peg 3, the task posed in 1883 (sans random moves) by mathematician Edouard Lucas, creator of the classic HT toy puzzle.   This method combines the commute time theorem for a random walk on a bidirectional graph with recursive applications of the delta-to-wye transformation for 3-phase circuits introduced in 1899 by A. E. Kennelly who, BTW, also discovered the ionosphere. Then we extend Alekseyev’s approach to obtain a recursion from n-1 to n disks of the MGF's of the RMHT hitting times for the problems he investigated.