Shortest Paths in the Tower of Hanoi Graph and Finite Automata

Source abstract

We present efficient algorithms for constructing a shortest path between two configurations in the Tower of Hanoi graph and for computing the length of the shortest path. The key element is a finite‐state machine which decides, after examining on the average only a small number of the largest discs (asymptotically, $\frac{63}{38} \approx 1.66$), whether the largest disc will be moved once or twice. This solves a problem raised by Andreas Hinz and results in a better understanding of how the shortest path is determined. Our algorithm for computing the length of the shortest path is typically about twice as fast as the existing algorithm. We also use our results to give a new derivation of the average distance $\frac{466}{885}$ between two random points on the Sierpiński gasket of unit side.