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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