• 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: 1594

Towers of Hanoi.

Extracts from this document...

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

...read more.

Middle

image15.pngimage16.png

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.

...read more.

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.

  1. 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.

...read more.

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

  1. Beyond Pythagoras

    I am now going to test it with one more example, to be sure that it is reliable. (2x3.5+1) �+(2*3.5(3.5+1)) �=(2x3.5(3.5+1)+1) � 8�+31.5�=32.5� 64+992.25=1056.25 Therefore 1056.25=1056.25 which is again correct as an equation must equal the same on both sides.

  2. Beyond Pythagoras

    I will find out the 7th term by using the nth term I have just worked out then check it against what I have in my sequence. So when n = 7 2 x 72 + 2 x 7 + 0 2 x 49 + 14 98 + 14 =

  1. Beyond Pythagoras

    = 2n + 4 b (medium side) = 4n + n2 + 3 c (largest side) = 4n + n2 + 5 a2 = (2n + 4) x (2n + 4) = 4n2 + 16 + 8n + 8n = 16 + 16n + 4n2 b2 = (4n + n2 + 3)

  2. Beyond Pythagoras ...

    divide by b/2 in brackets. So it will become 10= 8+ (8+2) divide b/2 in the brackets 102=82+62 Mahmoud Elsherif Beyond Pythagoras P.13 100=64+36 Then you times the bases 3,4,5 by three to get a=b+(b+3) So it will become 15=12+(12+3)

  1. Was Maths invented or discovered?

    This fact proves that maths was always there. Mathematics comes as a part of our thinking. Maths is discovered because young children learn how to add with no communication therefore it is learnt by reasoning and therefore a discovery. On the other hand there are inventions in the world of Maths.

  2. Research on Pythagoras and his work.

    According to Porphyry ([12] and [13]) Pythagoras was refused admission to all the temples except the one at Diospolis where he was accepted into the priesthood after completing the rites necessary for admission. Both men and women were permitted to become members of the Society, in fact several later women Pythagoreans became famous philosophers.

  1. Beyond Pythagoras.

    So, I will now write the answers for 4n�. This coursework from www.studentcentral.co.uk (http://www.studentcentral.co.uk/coursework/essays/2255.html) Reproduction or retransmission in whole or in part expressly prohibited 4n� work for the first term, but, it then collapses after this, as the difference between 4n� gets larger, the thing you notice is that the difference in the 2nd term between 4n� and the middle side is the middle side for the term before.

  2. The Die Investigation.

    P(A winning) + P(B winning) + P(C winning) = x/x 3/x + 5/x + 5/x =x/x *x) x = 3 + 5 + 5 x = 13 Therefore: P(A winning) = 3/13 P(B winning) = 5/13 P(C winning) = 5/13 The next question was to know how many goes should it be before someone wins.

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