This is an overview of some of the work I would like to pursue with undergraduates

There are several still-open questions involving the Towers of Hanoi puzzle. The valid positions of a Towers of Hanoi puzzle can be placed in a graph which turns out to be a Sierpinski triangle! Graph adjacencies represent states having a distance of one move. It is surprisingly simple (and fast) to compute the distance between any two valid positions of a Towers of Hanoi. However, this calculation involves computing two possible scenarios. Either the largest disc is moved exactly once (this is the most common arrangment), or the largest disc is moved twice. Additonal progress on this open problem was made by Dan Romik in roughly 2003.

Distances from a given location within a Sierpinski triangle.

Green points have two minimal paths to the red point. Blue points have the property that the minimal path to the red point requires moving the largest disc twice in the corresponding Hanoi problem.

This is similar, but for a Hanoi puzzle with more discs. Points are colored blue if the largest disc which was moved at all moved twice.