# Towers of Hanoi.

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Introduction

Year 10 GCSE coursework

## Towers of Hanoi

## Introduction

The aim of this piece of coursework is to complete different investigations. The name of these investigations is the Towers Of Hanoi. I will need to be patient and enthusiastic to complete these testing challenges. Basically I have 4 discs of decreasing radii and 3 towers named A, B and C. I am allowed to move only one disc at a time and I cannot place a larger disc on top of a smaller disc. I have to complete the challenges within a certain amount of goes. I will do 6 investigations using 1, 2, 3, 4, 5 and 6 discs. After I have completed these investigations I will compare them and try to find patterns etc. I will be required to show diagrams, graphs, tables of results and rules. I will also include a conclusion.

### Investigating some challenges

Now I am going to show my the 6 investigations and try to find patterns and rules afterwards

### Investigation 1

In

Middle

11) 12)

13) 14)

15)

Moves: 1-B

2-C

1-C

3-B

1-A

2-B

1-B

4-C

1-C

2-A

1-A

3-C

1-B

2-C

1-C

As I confirmed, it is possible to complete this task in a minimum of 15 moves.

### Investigation 2

In my second investigation I am going to try to successfully move one disc from the start (A) to the finish (B or C).

### Prediction

I predict that it will take me one move to get from start to finish as there’s only one disc and so I can move it anywhere I want in one move as there are no other discs.

This is the position I am going to start from.

1)

Moves: 1-C

My prediction was correct and I now can confirm it takes me 1 move to successfully move 1 disc from start to finish.

### Investigation 3

Now I am going to try to successfully move 2 discs from start (A) to finish (B or C) in the least number of moves possible.

This is the position I am going to start my challenge form.

1) 2)

3)

Moves: 1-B

2-C

1-C

My challenge was successful and I completed it in 3 moves.

### Investigation 4

Now in my fourth investigation I am going to attempt to move 3 discs from start (A) to finish (B or C) in the minimum amount of moves possible.

Conclusion

2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 – 1 = 255

2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 –1 = 511

As you can see all these sums work and I could carry on using the formula.

- Term to term rule:

This is a rule which enables me to find the amount of moves required for any number of discs. The rule is 2n+1. All I do is double the last term, For example: -

2 x 0 +1 = 1

2 x 1 +1 = 3

2 x 3 +1 = 7

2 x 7 +1 = 15

2 x 15 +1 = 31

2 x 31 +1 = 63

2 x 63 +1 = 127

2 x 127 +1 = 255

2 x 255 +1 = 511

2 x 511 +1 = 1023

As you can see the rule works well and so is very helpful in finding the next trem along. All you do is double the last term like double the 3 and +1 you get 7 and then you do the same again.

Conclusion

What I discovered is that there are simple ways of solving these investigations. These patterns and rules were really helpful in the end for my work as when I was trying to crack one of the puzzles I knew how many moves I had to do it in which help. I knew this due to the patterns I found and the rules. The 2 rules I found were the 2n-1 rule, which is a position to term rule, and the other rule, which is the term-to-term rule, is 2n+1. Overall I found that finding the rules and patterns came quite easily in the end as the patterns built up as I done more work.

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

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