I was not very satisfied with my earlier solution to DWW's ToH problem, and undertook a thorough investigation, hoping to explain more directly just why the answers come out as they do. I shan't take up bandwidth here with the results --- anybody wanting more detail is free to email for the relevant page of acrobat --- but a couple of things emerged which might interest people. Let the discs be indexed from 1 to n in order of decreasing diameter [apologies to DWW for previously dismissing this convention], and the pins 0, -1, -2 in order initial, intermediate, final. Given the move number in binary notation, there is an efficient algorithm to determine the index k of the disc/pin being shifted, l where it came off from, m where it goes on to. This can be reversed to enumerate all such moves, and in particular to evaluate their cardinal h(k,l,m). The construction hinges on an unexpected property of configurations along the solution path, novel to me, and apparently lacking any obvious combinatorial reason: For disc k to move to/from j < k-1, it is necessary that discs j+2, ..., k-2 occur in consecutive pairs sharing the same pin. [It can be easily established that the move must be from/to k-1.] Example: after move 28468 of the solution for n = 16 discs, the configuration is pin -0 discs 1 4 9 10 13 pin -1 discs 2 3 14 pin -2 discs 5 6 7 8 11 12 15 16 Disc k = 14 has just shifted from l = 13 to m = 3 = j; discs (5,6), (7,8), (9,10), (11,12) occur paired on pin -2,-2,0,-2 resp. By counting all configurations in which these are paired and later discs are on the unused pin, after a little tinkering with indices neighbouring k and j, we can further calculate that the number of moves with these values of k,l,m equals 2^((k+j-3)/2) = 128. Fred Lunnon [10/06/06]