A Tale of Two Puzzles: Towers of Hanoi and Spin-Out

Once upon a time, there were two puzzles. One was the Towers of Hanoi invented or introduced by Eduardo Lucas in 1883. The other was Spin-Out patented by William Keister in 1972. There are many stories about these puzzles. Some of these stories hint or claim that these puzzles have an intimate relationship with the Gray codes invented by Frank Gray in 1947. Here, we wish to show how these puzzles can be generalized and crossed to give puzzles for every base and for every number of pieces. The Gray relationship will become clearer when we describe the graphs associated with the puzzles and the graph labelings induced by the puzzles. These labelings will have the Gray property in the appropriate base. Counter to claims that Gray counting is needed to solve these puzzles, we describe counting algorithms which solve these puzzles using a standard binary counter. We also give recursive and iterative algorithms for these puzzles.