TITLE: Can someone intuitively explain the towers of Hanoi and how a proof by induction can be used?
QUESTION [5 upvotes]: I'm having trouble understanding precisely how the proof by induction is used to show that it takes at most $2^n-1$ moves to get all the disks from one peg to another.  Are there any helpful sources or particular insight that you have that may help me, and others, understand this problem more intuitively?

REPLY [2 votes]: The proof goes as follows:  for one disk, it takes $1=2^1-1$ move.  Assume for up to $n$ disks it takes $2^n-1$ moves.  Then for $n+1$ disks, you first have to move the top $n$ to an empty peg, taking $2^n-1$ moves.  Next, you move disk $n+1$ to the other empty peg, taking $1$ move.  Third, you move the tower of $n$ disks from where it is onto disk $n+1$, again taking $2^n-1$ moves.  The total number of moves is $(2^n-1)+1+(2^n-1)=2^{n+1}-1$