Tower of Hanoi

The Tower of Hanoi puzzle was invented by the French mathematician Edouard Lucas in 1883. We are given a tower of eight disks (initially four in the applet below), initially stacked in increasing size on one of three pegs. The objective is to transfer the entire tower to one of the other pegs (the rightmost one in the applet below), moving only one disk at a time and never a larger one onto a smaller.

The puzzle is well known to students of Computer Science since it appears in virtually any introductory text on data structures or algorithms. Its solution touches on two important topics discussed later on:

recursive functions and stacks
recurrence relations

The applet has several controls that allow one to select the number of disks and observe the solution in a Fast or Slow manner. To solve the puzzle drag disks from one peg to another following the rules. You can drop a disk on to a peg when its center is sufficiently close to the center of the peg. The applet expects you to move disks from the leftmost peg to the rightmost peg.

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Towers of Hanoi puzzle

What if applet does not run?

Recursive solution

Let call the three pegs Src (Source), Aux (Auxiliary) and Dst (Destination). To better understand and appreciate the following solution you should try solving the puzzle for small number of disks, say, 2,3, and, perhaps, 4. However one solves the problem, sooner or later the bottom disk will have to be moved from Src to Dst. At this point in time all the remaining disks will have to be stacked in decreasing size order on Aux. After moving the bottom disk from Src to Dst these disks will have to be moved from Aux to Dst. Therefore, for a given number N of disks, the problem appears to be solved if we know how to accomplish the following tasks:

Move the top N - 1 disks from Src to Aux (using Dst as an intermediary peg)
Move the bottom disk from Src to Dst
Move N - 1 disks from Aux to Dst (using Src as an intermediary peg)

Assume there is a function Solve with four arguments - number of disks and three pegs (source, intermediary and destination - in this order). Then the body of the function might look like


Solve(N, Src, Aux, Dst)
    if N is 0 
      exit
    else
      Solve(N - 1, Src, Dst, Aux)
      Move from Src to Dst
      Solve(N - 1, Aux, Src, Dst)

This actually serves as the definition of the function Solve. The function is recursive in that it calls itself repeatedly with decreasing values of N until a terminating condition (in our case N = 0) has been met. To me the sheer simplicity of the solution is breathtaking. For N = 3 it translates into

Move from Src to Dst
Move from Src to Aux
Move from Dst to Aux
Move from Src to Dst
Move from Aux to Src
Move from Aux to Dst
Move from Src to Dst

Of course "Move" means moving the topmost disk. For N = 4 we get the following sequence

Move from Src to Aux
Move from Src to Dst
Move from Aux to Dst
Move from Src to Aux
Move from Dst to Src
Move from Dst to Aux
Move from Src to Aux
Move from Src to Dst
Move from Aux to Dst
Move from Aux to Src
Move from Dst to Src
Move from Aux to Dst
Move from Src to Aux
Move from Src to Dst
Move from Aux to Dst

Recurrence relations

Let T_N be the minimum number of moves needed to solve the puzzle with N disks. From the previous section T₃ = 7 and T₄ = 15. One can easily convince oneself that T₂ = 3 and T₁ = 1. A trained mathematician would also note that T₀ = 0. Now let us try to derive a general formula.

The recursive solution above involves moving twice (N - 1) disks from one peg to another and making one additional move in between. It then follows that

T_N ≤ T_N-1 + 1 + T_N-1 = 2T_N-1 + 1

The inequality suggests that it might be possible to move N disks with fewer than 2T_N-1 + 1 moves. Which is actually not the case. Indeed, when the time comes to move the bottom disk, (N - 1) smaller disks will have been moved from Src to Aux in at least T_N-1 moves. Since we are trying to use as few steps as possible, we may assume that that portion of the task took exactly T_N-1 moves. It takes just one move to move the biggest disk from Src to Dst. One then needs exactly T_N-1 more steps to finish the task. Therefore the minimum number of moves needed to solve the puzzle with N disks equals T_N-1 + 1 + T_N-1 = 2T_N-1 + 1 moves.

In other words,

T_N = 2T_N-1 + 1

Thus we can define the quantity T_N as

T₀ = 0
T_N = 2T_N-1 + 1 for N > 0

We may compute T₁ = 2T₀ + 1 = 1, T₂ = 2T₁ + 1= 3, T₃ = 2T₂ + 1 = 7 and so on sequentially.

The above expression is known as a recurrence relation which, as you might have noticed, is but a recursive function. T_N is defined in terms of only one of its preceding values. Other recurrence relations may be more complicated, for example, f(N) = 2f(N - 1) + 3f(N - 2). Recurrence relations appear under various guises in numerous branches of Mathematics and applications.

Returning to the definition of T_N, define S_N = T_N + 1. Then S₀ = 1 and S_N = T_N + 1 = (2T_N-1 + 1) + 1 = 2T_N-1 + 2 = 2(T_N-1 + 1) = 2S_N-1. Which is to say that S_N could be defined as

S₀ = 1
S_N = 2S_N-1 for N > 0

The latter is solved easily in the closed (non-recurrent) form S_N=2^N. Wherefrom

T_N = 2^N - 1 for N ≥ 0.

Remark 1

The latest implementation of the applet sports a "Suggest move" button that exploits an algorithm devised by Romek Zylla who graciously put up on the Web an explanation of his algorithm. The algorithm actually provides another, a non-recursive solution to the puzzle.

Remark 2

There are quite a few variations on the puzzle. For example, you may want to experiment with its bicolor or 3 colors versions.

Reference

A. Beck, M. N. Bleicher, D. W. Crowe, Excursions into Mathematics, A K Peters, 2000

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