15925 hanoi towers

8/2/2019 15925 Hanoi Towers

1/33

The Towers of HanoiThe Towers of Hanoi


2/33

A LegendA Legend

Legend has it that there were three diamond needles setLegend has it that there were three diamond needles set

into the floor of the temple of Brahma in Hanoi.into the floor of the temple of Brahma in Hanoi.

Stacked upon the leftmost needle were 64 golden disks,Stacked upon the leftmost needle were 64 golden disks,each a different size, stacked in concentric order:each a different size, stacked in concentric order:


3/33

A Legend (A Legend (CtdCtd))

The priests were to transfer the disks from the first needleThe priests were to transfer the disks from the first needle

to the second needle, using the third as necessary.to the second needle, using the third as necessary.

But they couldBut they could only moveonly moveone disk at a timeone disk at a time, and could, and could

never put a larger disk on top of a smaller onenever put a larger disk on top of a smaller one..

When they completed this task,When they completed this task, the world would endthe world would end!!


4/33

To IllustrateTo Illustrate

For simplicity, suppose there were just 3 disks, and wellFor simplicity, suppose there were just 3 disks, and well

refer to the three needles as A, B, and C...refer to the three needles as A, B, and C...

Since we can only move one disk at a time, we move theSince we can only move one disk at a time, we move the

top disk from A to B.top disk from A to B.


5/33

ExampleExample



We then move the top disk from A to C.We then move the top disk from A to C.


6/33

Example (Example (CtdCtd))



We then move the top disk from B to C.We then move the top disk from B to C.


7/33




We then move the top disk from A to B.We then move the top disk from A to B.


8/33




We then move the top disk from C to A.We then move the top disk from C to A.


9/33




We then move the top disk from C to B.We then move the top disk from C to B.


10/33




We then move the top disk from A to B.We then move the top disk from A to B.


11/33




and were done!and were done!

The problem gets more difficult as the number of disksThe problem gets more difficult as the number of disks

increases...increases...


12/33

Our ProblemOur Problem

Todays problem is to write a program that generates theTodays problem is to write a program that generates the

instructions for the priests to follow in moving the disks.instructions for the priests to follow in moving the disks.

While quite difficult to solve iteratively, this problem hasWhile quite difficult to solve iteratively, this problem has

a simple and eleganta simple and elegant recursiverecursivesolution.solution.


13/33

AnalysisAnalysis

For flexibility, lets allow the user to enter the number ofFor flexibility, lets allow the user to enter the number of

disks for which they wish a set of instructions:disks for which they wish a set of instructions:

/* hanoi.cpp* ...*/

void Move(int n, char src, char dest, char aux);

int main(){

cout numDisks;

Move(numDisks, A, B, C);}


14/33

Analysis (Ctd)Analysis (Ctd)

Our task, then is to write function Move() that does allOur task, then is to write function Move() that does all

the work:the work:

/* hanoi.cpp* ...*/

void Move(int n, char src, char dest, char aux);

int main(){

cout numDisks;

Move(numDisks, A, B, C);}


15/33

DesignDesign

Basis: What is an instance of the problem that is trivial?Basis: What is an instance of the problem that is trivial?

n == 1n == 1

Since this base case could occur when the disk is on anySince this base case could occur when the disk is on any

needle, we simply output the instruction to move theneedle, we simply output the instruction to move the

top disk fromtop disk from srcsrctoto destdest..


16/33

DesignDesign

Basis: What is an instance of the problem that is trivial?Basis: What is an instance of the problem that is trivial?

n == 1n == 1

Since this base case could occur when the disk is on anySince this base case could occur when the disk is on any

needle, we simply output the instruction to move theneedle, we simply output the instruction to move the

top disk fromtop disk from srcsrctoto destdest..


17/33

Design (Design (CtdCtd))

Induction Step: n > 1Induction Step: n > 1

How can recursion help us out?How can recursion help us out?

a.a. RecursivelyRecursively move n-1 disks frommove n-1 disks from srcsrctoto auxaux..


18/33




b. Move the one remaining disk fromb. Move the one remaining disk from srcsrctoto destdest..


19/33




c.c. RecursivelyRecursivelymove n-1 disks frommove n-1 disks from auxauxtoto destdest......


20/33




d. Were done!d. Were done!


21/33

AlgorithmAlgorithm

We can combine these steps into the following algorithm:We can combine these steps into the following algorithm:

0.0. ReceiveReceive n, src, dest, auxn, src, dest, aux..

1.1. IfIfnn> 1:> 1:

a. Move(a. Move(n-1, src, aux, destn-1, src, aux, dest););

b. Move(1,b. Move(1, src, dest, auxsrc, dest, aux););

c. Move(c. Move(n-1, aux, dest, srcn-1, aux, dest, src););

ElseElse

Display Move the top disk from ,Display Move the top disk from , srcsrc, to ,, to , destdest..

End if.End if.


22/33

CodingCoding

// ...

void Move(int n, char src, char dest, char aux){if (n > 1){

Move(n-1, src, aux, dest);Move(1, src, dest, aux);Move(n-1, aux, dest, src);

}elsecout


23/33

[email protected]

Recursion tree:

The order of recursive calls that results from

solveTowers(3,A,B,C)
mailto:[email protected]:[email protected]


24/33

[email protected] AA BB CC

AA BB CC

AA BB CC

Figure 2.21aFigure 2.21a

Box trace ofsolveTowers(3, A, B, C)

AA BB CC


25/33

[email protected]

Figure 2.21bFigure 2.21b


AA BB CC

AA BB CC

AA BB CC

AA BB CC


26/33

[email protected]

Figure 2.21cFigure 2.21c


AA BB CC

AA BB CC

AA BB CC

AA BB CC


27/33

[email protected]

Figure 2.21dFigure 2.21d


AA BB CC


28/33

[email protected]

Figure 2.21eFigure 2.21e



29/33

[email protected]

TestingTestingThe Hanoi Towers

Enter how many disks: 1Move the top disk from A to B


30/33

[email protected]

Testing (Testing (CtdCtd))The Hanoi Towers

Enter how many disks: 2Move the top disk from A to CMove the top disk from A to BMove the top disk from C to B


31/33

[email protected]


Enter how many disks: 3Move the top disk from A to BMove the top disk from A to CMove the top disk from B to CMove the top disk from A to B

Move the top disk from C to AMove the top disk from C to BMove the top disk from A to B


32/33

[email protected]


Enter how many disks: 4

move a disk from needle A to needle B

move a disk from needle C to needle B

move a disk from needle A to needle C

move a disk from needle B to needle Amove a disk from needle B to needle C




move a disk from needle C to needle A

move a disk from needle B to needle Amove a disk from needle C to needle B





33/33

adams@calvin edu

AnalysisAnalysis

Lets see how many moves it takes to solve this problem,Lets see how many moves it takes to solve this problem,as a function ofas a function ofnn, the number of disks to be moved., the number of disks to be moved.

nn Number of disk-moves requiredNumber of disk-moves required

11 1122 33

33 77

44 1515

55 3131

......

ii 22ii-1-1

6464 226464 1 (a big number)1 (a big number)

15925 hanoi towers

Documents