## Iterative Towers of Hanoi

In this last exercise of the series, we will implement an iterative algorithm for the base problem of the Towers of Hanoi (one stack, 3 pegs, no extra movement restriction).

This algorithm is actually very simple: On odd moves, the smallest disk is moved in a given direction (either increasing 0->1->2->0 or decreasing 2-<1-<0-<2) while on even moves, the only possible move that does not imply the smallest disk is made. You stop as soon as the stack is rebuilt on another location.

The function that you now need to write takes two parameters: the initial position of the smallest disk (i.e., the peg initially containing the stack) and a boolean indicating whether the smallest disk should move in the increasing order or not.

The simplicity of this algorithm is actually almost deceiving. One could wonder on the interest of the recursive algorithms when such a simple iterative algorithm exists. My personal feeling is that this solution is nice to execute, but almost impossible to devise in the first place (I even suspect that the authors built this iterative solution from observations of the recursive execution)...

Predicting the effect of this function is also difficult: When the small disk moves in the increasing order, the stack is reconstructed on the right peg with 5 disks but on the left with 6 disks.

An interesting question is whether such simple iterative algorithms exist for the other variation of the problem. Some were given in the literature (e.g. for the cyclic variation). I would happily integrate your solutions to the PLM, especially if you can hint (without spoiling) the solution to guide the next ones.