This also applies to more complicated things than bouncing around a rectangle. For instance, solving a Towers of Hanoi puzzle, where the i'th step in a solution

This also applies to more complicated things than bouncing around a rectangle. For instance, solving a Towers of Hanoi puzzle, where the i'th step in a solution is to move a disc from peg (i&i-1)%3 to peg ((i|i-1)+1)%3). (Note that for even #discs, peg 1 is the destination). Example for 3 discs: i i-1 i&i-1 from i|i-1 to 1 0 0 0 1 2 2 1 0 0 3 1 3 2 2 2 3 1 4 3 0 0 7 2 5 4 4 1 5 0 6 5 4 1 7 2 7 6 6 0 7 2