Towers of Hanoi

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Towers of Hanoi
The Towers of Hanoi is a well known puzzle game publicized in 1883 by Édouard Lucas, a French mathematician. According to the legend brought forth by Lucas, priests of the Hindu god Brahma were instructed to move 64 golden disks from one of 3 poles to another, and when they completed it the world would end.
Towers of Hanoi
Field: Other
Created By: WikiBooks User GeniXPro

Contents

Basic Description

The standard game begins with 3 poles, one of which (the starting pole) has some natural number, k, disks placed around it in increasing size from top to bottom, while the other poles start out empty.
Goal: Move all k disks from the start pole (typically the leftmost) to the specified end pole (typically the rightmost), following these rules:
  • Only one disk may be moved at one time.
  • Each move consists of moving the top disk of one rod onto the top of another rod.
  • Each disk must be placed on a larger disk.

To fully understand the end of the world legend, one must understand the relationship between the number of disks and the amount of moves. As it turns out, this relationship is exponential; if the priests move one disk per second, then they would need 18,446,744,073,709,551,615 seconds, or 584.54 billion years, to complete the game. Given that this is about 58.5 times the life span of the Sun, the legend probably overestimates the amount of time we have until the world ends (unless we can leave the Earth before the Sun expands too much).

Play the Game

You can play the game Towers of Hanoi right here! This applet will help you understand how the game works.

If you can see this message, you do not have the Java software required to view the applet.

Example of Solving the Game

Solving Towers of Hanoi with 4 disks.

Image:Hanoi.gif

A More Mathematical Explanation

The Recursive Solution

This game becomes more interesting when we try to figure out the quickest [...]

The Recursive Solution

This game becomes more interesting when we try to figure out the quickest solution to any Towers of Hanoi game. We will first reason through the ideal minimum amount of steps to solve a game with k-disks, then provide an algorithm that requires the same amount of moves as this ideal minimum. Define T_k to be the minimum amount of moves needed for k disks. It is obvious that:

T_0=0 and T_1=1

Assume that we know T_{k-1} is true. Note that to move the kth(bottom) disk, we must first move the top k-1 disks. In addition, we must move the kth disk at least (and hopefully only) once. Finally, the top k-1 disks must be placed on top of the kth disk at some point to complete the game. Thus, the ideal recursive function for T_k is:

T_k=T_{k-1}+1+T_{k-1}=2T_{k-1}+1


Now we will provide an algorithm that requires the same amount of steps as T_k. Since T_0 and T_1 do not change, we can assume that the (k-1)th game is solvable. One algorithm to solve the kth game is as follows:

1. Move the top k-1 disks to Pole B (assuming we start on Pole A and end on Pole C).
2. Move the kth disk to Pole C.
3. Move the k-1 disks on Pole B to Pole C, on top of the kth disk.

The total steps for this are:

T_{k-1}+1+T_{k-1}=2T_{k-1}+1=T_k

Thus our ideal minimum function is a plausible solution to any Towers of Hanoi game.

The Explicit Formula

We will show by induction that the explicit formula is

T_k=2^k-1\text{, for }k\in\mathbb{Z}_+

The base case is satisfied:

T_0=2^0-1=1-1=0 as desired.

Assume the (k-1)th case holds. Then

T_k=2T_{k-1}+1=2(2^{k-1}-1)+1=2(2^{k-1})-2+1=2^k-1

Thus we have shown the non-recursive equation holds for any Towers of Hanoi game with k disks.

Back to the Legend

Now that we have the equation to determine the amount of moves needed for any number of disks, we can verify the numbers listed in the Basic Description. A Towers of Hanoi game with 64 disks will take:

T_{64}=2^{64}-1=18,446,744,073,709,551,615\text{ moves}

If the priests move 1 disk per second, then they will need:

18,446,744,073,709,551,615\text{ seconds}=\left(\frac{1\text{ minute}}{60\text{ seconds}}\right)=\left(\frac{1\text{ hour}}{60\text{ minutes}}\right)=\left(\frac{1\text{ day}}{24\text{ hours}}\right)=\left(\frac{1\text{ year}}{365.25\text{ days}}\right)\approx584.54\text{ billion years}

Assuming that the Sun has a life span of 10 billion years, the time predicted by the legend is 585.54\div10\approx58.5 times longer than the Sun's lifespan.


Why It's Interesting

Towers of Hanoi is a popular example in introductory computer science courses; programming it is relatively simple but not trivial. The recursive nature of the game teaches recursion, while the simple interface of the game introduces basic graphics processing. In addition, the game's exponential solution shows why computer scientists dread exponential time; it takes far long to compute. Assuming the average computer can do 100 million operations per minute, then solving Towers of Hanoi with 64 disks would take the computer 5845.42 years to complete.


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