There's a very good book by Hinz, Klávžar, Milutinović and Petr, called "Towers of Hanoi - Myths and Maths".
It collects all sorts of theorems and perspectives on the puzzle, and plenty of variants.
Published by Springer: http://www.springer.com/gb/book/9783034802369
One variant that's easy to describe but hard to play is Twin Towers: get two sets, and each move consists of nominating two towers and making the valid move between them on both sets simultaneously.
Investigated by Zoran Šunić: http://read.somethingorotherwhatever.com/entry/Sunic2011
A Mastodon instance for maths people. The kind of people who make \(\pi z^2 \times a\) jokes.
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