Sometime in the mid-1990's, Paul Stockmeyer proposed a variation of the Reve's puzzle (Tower of Hanoi with 4 pins) which I propose to christen "four-star": one of the intermediate pins is distinguished, and every move is obliged to involve this pin either as origin or as destination. [ See "Variations on the Four-Post Tower of Hanoi Puzzle" at http://www.cs.wm.edu/~pkstoc/toh.html ] The analysis proceeds much as for the Reve's puzzle. To transfer n discs from origin pin to destination pin: (1) transfer n-k smaller discs from origin to (other) intermediate; (2) transfer k larger discs from origin to destination; (3) transfer n-k smaller discs from intermediate to destination; To choose the optimum value for k, Stockmeyer constructs an integer sequence a_n by arranging the set {2^i 3^j} in ascending order of magnitude, then sets k = k_n = 1 + [log(a_n)/log3]; e.g. (suitably fudged at n = 0) n = 0 1 2 3 4 5 6 7 8 9 10 11 12 a_n = ? 1 2 3 4 6 8 9 12 16 18 24 27 k_n = 0 1 1 2 2 2 2 3 3 3 3 3 4 The number of moves is then optimised at 2 (a_0 + ... + a_n), However, a much simpler algorithm would be simply to set k_n = [sqrt(2 n log2/log3) + 0.1809] ?? giving the same result for n < 1737 [which should satisfy the most enthusiastic apocryphal Brahmin!] It's not hard to see why such a formula might provide a good approximation; what I find surprising is that it fails only after such a protracted run. Fred Lunnon