On the Frame--Stewart Conjecture about the Towers of Hanoi

The multipeg Towers of Hanoi problem consists of k pegs mounted on a board together with n disks of different sizes. Initially these disks are placed on one peg in the order of their size, with the largest at the bottom. The rules of the problem allow disks to be moved one at a time from one peg to another as long as a disk is never placed on top of a smaller disk. The goal of the problem is to transfer all the disks to another peg with the minimum number of moves, denoted by H(n, k). An easy recursive argument shows that H(n, 3) = 2n−1. However, the problem of computing the exact value of H(n, k) for k ≥ 4 has been open since 1939, and in particular, the special case of H(n, 4) has been open since 1907. In 1941, Frame and Stewart each gave an algorithm to solve the Towers of Hanoi problem based on an unproved assumption. The Frame–Stewart number, denoted by FS(n, k), is the number of moves needed to solve the Towers of Hanoi problem using the “presumed optimal” Frame–Stewart algorithm. Since then, proving the Frame–Stewart conjecture FS(n, k) = H(n, k) has become a notorious open problem. In this paper, we prove that FS(n, k) and H(n, k) both have the same order of magnitude of 2(1±o(1))(n(k−2)!) 1/(k−2) . This provides the strongest evidence so far to support the Frame–Stewart conjecture.