Question 126
Michelle has a word with $2^{n}$ letters, where a word can consist of letters from any alphabet. Michelle performs a switcheroo on the word as follows: for each $k = 0, 1, \ldots, n-1$, she switches the first $2^{k}$ letters of the word with the next $2^{k}$ letters of the word. For example, for $n = 3$, Michelle changes
$$
A\ B\ C\ D\ E\ F\ G\ H \rightarrow B\ A\ C\ D\ E\ F\ G\ H \rightarrow C\ D\ B\ A\ E\ F\ G\ H \rightarrow E\ F\ G\ H\ C\ D\ B\ A
$$
in one switcheroo.
In terms of $n$, what is the minimum positive integer $m$ such that after Michelle performs the switcheroo operation $m$ times on any word of length $2^{n}$, she will receive her original word?
$$
A\ B\ C\ D\ E\ F\ G\ H \rightarrow B\ A\ C\ D\ E\ F\ G\ H \rightarrow C\ D\ B\ A\ E\ F\ G\ H \rightarrow E\ F\ G\ H\ C\ D\ B\ A
$$
in one switcheroo.
In terms of $n$, what is the minimum positive integer $m$ such that after Michelle performs the switcheroo operation $m$ times on any word of length $2^{n}$, she will receive her original word?