Twitter proof: the Tower of Hanoi


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In this post we prove what the minimum number of moves to solve the problem of the Tower of Hanoi is!

Claim: let $T(n)$ denote the number of moves it takes to solve the Tower of Hanoi with $n$ disks; then $T(n) = 2^{n}-1$.

Twitter proof: note that to solve the problem with $n$ disks, we first have to move the top $n-1$ disks to one of the two poles, move the bottom disk (the bigger one) to the remaining pole, and then move the top $n-1$ disks to the top of the bigger disk. Each time we move the top $n-1$ disks to another pole we must take, at least, $T(n-1)$ moves (by definition of $T$) hence we clearly have $T(n) = 2T(n-1) + 1$. Just notice that $B(n) = 2^n - 1$ satisfies the recurrence relation and that $T(0) = B(0) = 0$.

If you are having trouble understanding what I mean by to solve the problem with $n$ disks, we first have to move the top $n-1$ disks to one of the two poles, move the bottom disk (the bigger one) to the remaining pole, and then move the top $n-1$ disks to the top of the bigger disk, check the following sequence of images that contains some of the intermediate steps to solve the Tower of Hanoi with $4$ disks:


First we move the top $3$ disks to the centre pole in $T(3)$ moves...


Then we change the bottom disk to the available pole in $1$ move...


Then we move the top $3$ disks again in $T(3)$ moves!

  - RGS

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