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 pegs.
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Notes
Such positions could instead be declared drawn and we discuss some variation in Sect. 7.
But, for future reference, we note that there is also another choice for cyclic scoring games, if one of the players repeatedly moves to more negative scores than the other player, then, even though the game might not terminate, one could define it as a loss for the ‘more negative’ player.
This algorithm produces an optimal path for four pegs (Bousch 2014) and is conjectured optimal for any \(k > 4\).
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We thank the anonymous referees for their comments and suggestions.
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The second author was partially supported by the Killam Trusts.
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Chappelon, J., Larsson, U. & Matsuura, A. Two-player Tower of Hanoi. Int J Game Theory 47, 463–486 (2018). https://doi.org/10.1007/s00182-017-0608-4
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DOI: https://doi.org/10.1007/s00182-017-0608-4