• 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
  4. 4
    4
  5. 5
    5
  6. 6
    6
  7. 7
    7
  8. 8
    8
  9. 9
    9
  10. 10
    10
  • Level: GCSE
  • Subject: Maths
  • Word count: 1626

Towers of Hanoi

Extracts from this document...

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.

...read more.

Middle

Move ORANGE from C to B Move BLUE from C to AMove ORANGE from B to AMove YELLOW from B to CMove ORANGE from A to CMove BLUE from A to BMove ORANGE from C to BMove GREEN from A to CMove ORANGE from B to AMove BLUE from B to CMove Orange from A to C

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

...read more.

Conclusion

n-1 + R n) –

             (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.  

...read more.

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

See related essaysSee related essays

Related GCSE Consecutive Numbers essays

  1. GCSE Maths Coursework - Maxi Product

    3+10+2 --> 3x10x2=60 (3,9,3)= 15 --> 3+9+3 --> 3x9x3 =81 (4,8,3)= 15 --> 4+8+3 --> 4x8x3 =96 (4,7,4)= 15 --> 4+7+4 --> 4x7x4 =112 (5,6,4)= 15 --> 5+6+4 --> 5x6x4 =120 (5,5,5)= 15 --> 5+5+5 --> 5x5x5 =125 I am now going to use decimal numbers as I have found the highest possible result in whole numbers.

  2. Investigate the Maxi Product of numbers

    7.2x7.8 56.16 15 7.3 and 7.7 7.3+7.7 7.3x7.7 56.21 15 7.4 and 7.6 7.4+7.6 7.4x7.6 56.24 15 7.5 and 7.5 7.5+7.5 7.5x7.5 56.25- Maxi Product 15 7 2/9 and 7 7/9 7 2/9+7 7/9 7 2/9x7 7/9 56.179 (3dp) 15 7 5/9 and 7 4/9 7 5/9+7 4/9 7 5/9x7 4/9 56.247 (3dp)

  1. In this investigation I will explore the relationship between a series of straight, non-parallel, ...

    is correct because there are 14 Open Regions in this diagram, and that is what I predicted. Step 2: I will list the number of cross-over points to investigate the pattern in the second sequence. The number of Cross -Over Points is given by: COP(n)

  2. Nth Term Investigation

    each cube has 8 because the symbol represents the corners and all cubes have 8 corners. The nth term is n= 8. For the third column ( ) the amount goes up in 8's because an extra symbol is needed 1 more time on each of the 8 sides.

  1. I am to conduct an investigation involving a number grid.

    - (x2 + 12x) = 20 48 - 28 = 20 The difference between the two numbers is 20 Box 2 24 25 26 29 30 31 34 35 36 [image024.gif] 24 x 36 = 864 34 x 26 = 884 884 - 864 = 20 The difference between the

  2. The Towers of Hanoi is an ancient mathematical game. The aim of this coursework ...

    +1= 2(7) + 1 = 15 5. For 5 discs, it takes 31 moves: 2(Un-1) +1= 2(15) + 1 = 31 To understand how this works, coding is needed to see how a disc moves individually. Coding should show me the patterns involved and then I should be able to justify my formula based on this.

  1. Towers of Hanoi.

    Number of discs 2 to the power of the number of discs 1 2 2 4 3 8 4 16 5 32 Then from looking at the results again, I noticed that the difference between the 'Smallest number moves for each disc' and the 'Difference between the number of moves' was only that of 1.

  2. Investigate calendars, and look for any patterns.

    by six days. I have also noticed that descending down (i.e. Columns) is always n = 7n - x; and that descending diagonally right gives n = 8n - x. 5. Studying relationships between adjacent numbers I drew a box around 4 numbers: Ex 5.1 Starting 2nd August 2 3

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