# Towers of Hanoi

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Introduction

Towers of Hanoi

## Introduction:

Towers of Hanoi is a puzzle/game in which you have to move a certain number of discs from pole a to pole c in the minimum amount of moves possible. There are a few certain rules you have to follow though:

1. You CANNOT move 2 discs at a time

2. You CANNOT place a smaller disc over a bigger disc.

Aim:

The main of the investigation is to enquire into the relationship between the number of discs and the minimum number of moves to complete. We also have to investigate the symmetry of where the disc’s go and move to the poles. Also number of moves made by the individual discs to be moved at each stage of the process. We also have to find a formula linking the minimum no of moves to the number of discs.

How I went about finding this data:

When we were told to investigate Towers of Hanoi we had to figure how many moves it would take to get to pole C in the least amount of moves for 4 discs. So I got a plain piece of paper and drew the three poles on it. Then I got 4 different coloured pieces of paper from the teacher and sat down and tried figuring it out. I also played the game on the schools computers.

Middle

Others formulas I noticed

Symmetry formula:

N=2C +1 Where N equals the number of moves and C equals the previous number of moves.

NUMBER OF DISCS | MIMUNUM NUMBER OF MOVES |

0 | 0 |

1 | 1 |

2 | 3 |

3 | 7 |

4 | 15 |

5 | 31 |

6 | 63 |

7 | 127 |

By looking at this table which I figured by using the other formula I found (look at next formula) you can see the formula in it already

E.G- for finding the minimum number of moves for 6 discs

2 x 31=62 62+1=63 63 is the Minimum number of moves for 6 discs.

You can also find this formula by looking at the Running commentary of any Disc number.

E.G-Running Commentary of 3 Discs

Move GREEN from A to C

Move YELLOW from A to B the colour sequence is exactly the same as

Move GREEN from C to B

Move RED from A to C

Move GREEN from B to A

Move YELLOW from B to C This one

Move GREEN from A to C

So all you have to do is

Conclusion

(R 0 + R 1 + R 2 + R 3 +………….R n-1)

The cross out the numbers means the cancelling of the numbers because they are they same.

Rsn-Sn= R n – R 0

Rsn-Sn= R n-1

Sn (R-1)= R n –1

So…

Sn=R n-1 or Sn=R n -1

1 R-1

E.G. Finding out the number of moves for 7 discs using G.P

Formula is: Sn= R n -1

1

Substitute the numbers in-

Sn = 2 7-1

1

2 7 = 128 128-1= 127 127/1=127

127= Minimum number of moves for 7 discs

Graphs

The graphs are on the graph paper on the next page.

All the graphs are Exponential graphs which means that they all go up in a

Curved line from steady rise to a dramatic and big rise. This shows a relationship between the axis on the graph.

Conclusion

Evaluation

To make this investigation better you could have also seen if their was a relationship between the Minimum number of moves and also what pole you finish on (instead of pole c try to make it pole b). The only real difficulties I faced was trying to figure out the Geometric Progression formula but after additional research I fully understood it and was able to find the formula.

If I were to do it again I would see if there was a formula to see if you just knew the amount of moves then how would you figure the number of discs. You would have to do it without trying to use the other formula. Although this investigation was sufficient enough to answer the aim.

This student written piece of work is one of many that can be found in our GCSE Consecutive Numbers section.

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