DISKS ON A CHESSBOARD

The puzzle in this note is similar to the one that was analyzed in [1]. After characterizing the configurations in our puzzle as representing cyclic permutations, we apply this result to the traveling salesman problem. We begin with an infinite "chessboard" covering the first quadrant. The cells of the board are labelled by integer coordinates (i, j) with i, j 2 O. Initially, a single "disk" is located in cell (O, O). The first step consists of replacing this disk by two disks, located in cells (1, O) and (O, 1), respectively. In general, if there are k disks on the board, a step will consist of removing some disk, say in cell (i, j), and placing two disks on the board, one disk located in cell (k, j) and the other disk located in cell (i, k).