Rules and practice of the Game of the TOWER OF HANOI


The base is placed horizontally; the pegs stand upright in the holes in the surface. Then, the eight disks are stacked in decreasing order from base to summit; this creates the Tower.

The game consists of moving this, by threading the disks on another peg, and by moving only one disk at a time, obeying the following rules:
I. -- After each move, the disks will all be stacked on one, two, or three pegs, in decreasing order from the base to the top.
II. -- The top disk may be lifted from one of the three stacks of disks, and placed on a peg that is empty.
III. -- The top disk may be lifted from one of the three stacks and placed on top of another stack, provided that the top disk on that stack is larger.


The Game easily teaches itself, in solving first the problem for 3, 4, and 5 disks.

The Game is always possible and requires double the time each time that an additional disk is placed on the tower. Anyone who knows how to solve the problem for eight disks, for example, moving the tower from peg number 1 to peg number 2, also knows how to solve it for nine disks. One first moves the eight top disks to peg number 3, then moves the ninth disk to peg number 2, and then brings the eight disks from peg number 3 to peg number 2. Now, in augmenting the tower by one disk, the number of moves is doubled, plus one, from the preceding game.

For a tower of two disks, three moves are required;
-three -- seven ---
-four -- fifteen ---
-5 -- 31 ---
-6 -- 63 ---
-7 -- 127 ---
-8 -- 255 ---
and so on.

At one move per second it requires more than four minutes to move a tower of eight disks.

Variations of the Game. -- One varies, to infinity, the conditions of the problem of the tower of Hanoi as follows. At the beginning, one stacks the disks, in any manner, on one, two, or all three pegs. It is necessary to reconstruct the Tower on one of the pegs designated in advance. For 64 disks, the number of initial arrangements is staggering; the number is more than 50 digits long.

For more details, consult the following work in the chapter on the Baguenaudier:


by Mr. Édouard Lucas,
professor of higher mathematics at the Lycée Saint-Louis

Two small volumes, in two colors
Paris, 1883, by GAUTHER-VILLARS,
printer of the Académie des Sciences and the Ecole Polytechnique
Quai des Augustins, 55

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Last updated 27 August 1998