Two-Player Tower of Hanoi

TL;DR

This paper extends the classical Tower of Hanoi puzzle to a two-player setting, analyzing winning strategies and score optimization on three heaps, bridging recreational mathematics and combinatorial game theory.

Contribution

It introduces and solves two-player versions of the Tower of Hanoi with classical winning conditions on three heaps, a novel extension of the traditional puzzle.

Findings

Characterization of winning strategies for two-player Tower of Hanoi

Solution of last-move and highest-score winning conditions on three heaps

Extension of classical puzzle to competitive game scenarios

Abstract

The Tower of Hanoi game is a classical puzzle in recreational mathematics (Lucas 1883) which also has a strong record in pure mathematics. In a borderland between these two areas we find the characterization of the minimal number of moves, which is $2^n-1$, to transfer a tower of $n$ disks. But there are also other variations to the game, involving for example real number weights on the moves of the disks. This gives rise to a similar type of problem, but where the final score seeks to be optimized. We study extensions of the one-player setting to two players, invoking classical winning conditions in combinatorial game theory such as the player who moves last wins, or the highest score wins. Here we solve both these winning conditions on three heaps.