There are two cases when the largest disk moves at least twice from in; the second
movement of the largest disk is to a peg j = i , or back to peg i . We will first treat the
first case, and then the second case. Without loss of generality, aiming for a contradiction,
assume that there is a shortest path from 0n to v moving the largest disk at least
twice—first to peg 1 and then to peg 2. Then we can write the path as a sequence of
steps: