![]() We represent moves by three actions $a_$) indicate progress (“ $ 1$”) or regress (“ $−1$”) in solving the classical puzzle (we add a game reset move when the classical puzzle is solved) and their base letters indicate whether the two disks to be lifted at the respective vertices of this edge are of matching (“ $x$”) or mismatching (“ $y$”) color. So the goal becomes how many moves does it take to transform $11\cdots 1$ to $33\cdots 3$. This uniquely represents a configuration because larger disks need to be below smaller disks. The $k$th entry of the string is which peg the $k$th smallest disk is on. The set of configurations of the game are represented by a string of $n$ $1$'s, $2$'s, or $3$'s. Let's stick with 3 pegs, and $n$-disks, it gets confusing with more pegs. I'll give a quick overview of how the model works. I might add V Nerkrashevych's book "Self-similar groups" or " From fractal groups to fractal sets" for the theory, though I forget off hand if the Hanoi towers game is in there. The language of finite automata is useful in understanding these groups Mangual did a fairly thorough rundown of references. I'd say this mostly falls under the label of geometric group theory, and is quite useful in the study of dynamical systems. The language which I have usually seen it written in is that of self-similar groups, which are automorphisms of rooted trees.
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