The space of legal configurations of the Towers of Hanoi puzzle with $n$ disks approximates Sierpinski's triangle.

There is a Hamiltonian path in the space of configurations which can be describes as forbidding both the transition $a\to b$ and undoing the previous step. This sweeps out the approximation to Sierpinski's triangle in the same way as the L-system (reflected from the one you show). As you do this for a puzzle with $n$ disks, you can ignore the smallest disk to get a similarly restricted path in the configurations of a puzzle with $n-1$ disks.

alt text http://upload.wikimedia.org/wikipedia/commons/thumb/d/da/Tower_of_Hanoi-3_Longest_Path.svg/500px-Tower_of_Hanoi-3_Longest_Path.svg.png

The space of legal configurations of the Towers of Hanoi puzzle with $n$ disks approximates Sierpinski's triangle.

There is a Hamiltonian path in the space of configurations which can be describes as forbidding both the transition $a\to b$ and undoing the previous step. This sweeps out the approximation to Sierpinski's triangle in the same way as the L-system (reflected from the one you show). As you do this for a puzzle with $n$ disks, you can ignore the smallest disk to get a similarly restricted path in the configurations of a puzzle with $n-1$ disks.