Shortest paths between regular states of the Tower of Hanoi

Abstract Two generalizations of the classical Tower of Hanoi problem 0, i.e., to transfer n discs from one peg to another by a series of legal moves, are considered: In 1-type problems the initial configuration is any regular distribution of the n discs among the three pegs, whereas problems of type 2 deal with arbitrary initial and final states. The interest in measuring the distance, i.e., in finding shortest paths between regular states was reinforced by experiments in cognitive psychology and artificial intelligence. The question has, however, caused much confusion in the literature. Setting out from the well-known solution for 0, some mathematical background is developed to construct simple algorithms that solve problems 1 and 2 correctly and return the minimum number of moves as well as the instructions for the first move of a shortest path.

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