Towers of Hanoi

3. Move GREEN from C to B

4. Move RED from A to C

5. Move GREEN from B to A

6. Move YELLOW from B to C

7. Move GREEN from A to C

Table of 3 discs

Running Commentary of 4 discs:

                 A                    B                 C

1. Move BLUE from A to B                 15.        Move Blue from B to C

2. Move GREEN from A to C

3. Move BLUE from B to C

4. Move Yellow from A to C

5. Move BLUE from C to A

6. Move GREEN from C to B

7. Move BLUE from A to B

8. Move RED from A to C

9. Move BLUE from B to C

10. Move GREEN from B to A

11. Move BLUE from C to A

12. Move Yellow from B to C

13. Move BLUE from A to B

14. Move GREEN from A to C

Tables of 4 discs:

Running Commentary of 5 Discs

1. Move ORANGE from A to C                        
2. Move BLUE from A to B
3. Move ORANGE from C to B
4. Move GREEN from A to C
5. Move ORANGE from B to A
6. Move BLUE from B to C
7. Move ORANGE from A to C
8. Move YELLOW from A to B
9. Move ORANGE from C to B
10. Move BLUE from C to A
11. Move ORANGE from B to A
12. Move GREEN from C to B
13. Move ORANGE from A to C
14. Move BLUE from A to B
15. Move ORANGE from C to B
16. Move RED from A to C
17. Move ORANGE from B to A
18. Move BLUE from B to C
19. Move ORANGE from A to C
20. Move GREEN from B to A
21. Move ORANGE from C to B
22. Move BLUE from C to A
23. Move ORANGE from B to A
24. Move YELLOW from B to C
25. Move ORANGE from A to C
26. Move BLUE from A to B
27. Move ORANGE from C to B
28. Move GREEN from A to C
29. Move ORANGE from B to A
30. Move BLUE from B to C
31. Move 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.

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 add them two together and then add the largest disc (the one in the middle of the sequence, in this case its RED) so the working out is:

 2 x 3 = 6                6 + 1 = 7                 7 =number of moves for 3 discs

Other Formula I found:

To figure out the task, I knew that there must be a formula to it, so we could figure out the minimum number of moves for a certain number of discs (without actually having to physically play the game on the computer or figuring it out by doing it on a sheet of paper) to complete the game.

The Formula I found was Number of moves = 2d –1.

Where D is the number of discs to the power of 2. I found this formula by instead of multiplying the number of moves by 2 for each disk for the last number of moves for each number of discs; you could do it as a power instead of multiplying it each time.  

The power is the number of discs because I tried by putting the number of discs as a power (I found that if I used 2(which is the ratio) to the power of the number of discs) then the answer came up to be 1 more than the minimum number of moves, so if you take away 1 you will get the right answer.

E.G – Minimum number of moves for 6 discs

2 to the power of 6 = 64 or 2 6                64-1= 63         

63= minimum number of moves for 6 discs

Geometric Progression

Our task was to investigate the minimum number of moves required to complete the Towers of Hanoi game with different amounts of discs.  We investigated the minimum number of moves needed for 3,4 and 5 discs, and then I attempted using geometric progression (G.P) I was able to prove that my rule works and predict the minimum number of moves needed for any mount of discs.

Working Out  

Sn = Number of Moves

N = Number of discs

I.e. for 4 discs:

R = ratio/multiple

                                     

Multiply (ratio) = 2        

So R = 2

So R N-1  as N – is always one extra to the term

Sn=(R 0 + R 1 + R 2 +……………R n-1)

RSn= R (R 0 + R 1 + R 2 +…………….R n-1)  

RSn= (R 1 + R 2 + R 3 +……………..R n-1 + R n)

RSn-Sn=(R 1 + R 2 + R 3 +…………….R 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.