• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month
Page
1. 1
1
2. 2
2
3. 3
3

# Towers of Hanoi

Extracts from this document...

Introduction

## Towers of Hanoi

I am going to investigate the Hanoi Towers. The objective of the game is to move the discs from position A to positions B or C in the minimum number of moves where in one go you are only allowed to move one disc. I will vary the number of discs and record my results. I will make predictions and look for patterns. I will ultimately aim to find a formula for the number of moves it takes to move the tower from A to B or C.

Suppose you just had one disk. Then the method for moving that one disk from the starting spindle to the ending spindle is simple: just move it. It takes one step :

Now suppose you had two disks.

Middle

I now know that it is possible to move a tower of 4 discs in a minimum of 15 moves. I will know find the least amount of moves required to complete the game when you start with a tower of 5 discs.
I now know that it is possible to move a tower of 5 discs in a minimum of 31 moves.

Conclusion

s

1

3

7

15

31

2     4    8     16

From the bottom row of differences I can see that the previous term doubled gives the next term in the sequence. Therefore I predict that the next difference would be 32 for a tower of 6 discs (see results table) giving the number of moves as 63 also I can see that the previous number of moves multiplied by 2 plus 1 gives the next number of moves e.g.

1 x 2 + 1 = 3
3 x 2 + 1 = 7
7 x 2 + 1 = 15
15 x 2 + 1 = 31
31 x 2 + 1 = 63

So I can now construct a formula:

The nth term = the previous term doubled, plus one

or

Tn = 2Tn-1 + 1

eg. 1 Tower 2 x 0 +1 = 1
2 Towers 2 x 1 +1 = 3
3 Towers 2 x 3 +1 = 7
4 Towers 2 x 7 +1 = 15
5 Towers 2 x 15 +1 = 31
6 Towers 2 x 31 +1 = 63
7 Towers 2 x 63 +1 = 127
8 Towers 2 x 127 +1 = 255
9 Towers 2 x 255 +1 = 511
10 Towers 2 x 511 +1 = 1023

This student written piece of work is one of many that can be found in our GCSE T-Total section.

## Found what you're looking for?

• Start learning 29% faster today
• 150,000+ documents available
• Just £6.99 a month

Not the one? Search for your essay title...
• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month

# Related GCSE T-Total essays

1. ## I am going to investigate how changing the number of tiles at the centre ...

I have drawn pattern number 7 and counted 36 border tiles, which confirms that my formula is correct. Total Tiles I am also going to investigate whether I can find a formula which give the total amount of tiles in my pattern, T, provided I have the pattern number.

2. ## 3 Step Stairs

+ ( 10s x u ) ) By using the same method as before, I can work out the formulas for the other positions of 3 step stairs. For this position, in 3 step stairs, the formula would be: n = 90 + ( ( s x r )

• Over 160,000 pieces
of student written work
• Annotated by
experienced teachers
• Ideas and feedback to
improve your own work