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Investigating Towers of Hanoi Mathematics coursework

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Introduction

Investigating Towers of Hanoi
Mathematics coursework

Surname: Thomas
First name: Jawhari


Candidate no: 7160
Centre no: 14368

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The legend


In an ancient city in a temple monks were given the task of moving a pile of 64 sacred disks from one location to another. The disks are fragile; only one can be carried at a time. A disk may not be placed on top of a smaller, less valuable disk. And, there is only one other location in the temple, (besides the original and destination locations) sacred enough that a pile of disks can be placed there. The monks start moving disks back and forth, between the original pile, the pile at the new location, and the intermediate location, always keeping the piles in order (largest on the bottom, smallest on the top). The legend is that, before the monks make the final move to complete the new pile in the new location, the temple will turn to rubble and the world will end.


The Game. 

There's a game based on this legend. You have collection of disks and three piles into which you can place them.

...read more.

Middle

  1. Pile of 1, 1 move = 1-c.
  2. Pile of 2, 3 moves = 1-b, 2-c, 1-c.
  3. Pile of 3, 7 moves = 1-c, 2-b, 1-b, 3-c, 1-a, 2-c, 1-c.  
  1. Pile of 4, 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.
  2. Pile of 5, 31 moves = 1-c, 2-b, 1-b, 3-c, 1-a, 2-c, 1-c, 4-b, 1-b, 2-a, 1-a, 3-b, 1-c, 2-b, 1-b, 5-c, 1-a, 2-c, 1-c, 3-d, 1-b, 2-a, 1-a, 4-c, 1-c, 2-b, 1-b, 3-c, 1-a, 2-c, 1-c.
  3. Pile of 6, 63 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, 5-b, 1-a, 2-b, 1-b, 3-a, 1-c, 2-a, 1-a, 4-b, 1-b, 2-c, 3-b, 1-a, 2-b, 1-b, 6-c, 1-3, 2-a, 1-a, 3-c, 1-b, 2-c, 1-c, 4-a, 1-a, 2-b, 1-b, 3-a, 1-c, 2-a, 1-a, 5-c, 1-b, 2-a, 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
  4. Pile of 7, 127 moves = 1-c, 2-b, 1-b, 3-c, 1-a, 2-c, 1-c, 4-b, 1-b, 2-a, 1-a, 3-b, 1-c, 2-b, 1-b, 5-c, 1-a, 2-c, 1-c, 3-a, 1-b, 2-a, 1-a, 4-c, 1-c, 2-b, 1-b, 3-c, 1-a, 2-c, 1-c, 6-b, 1-b, 2-a, 1-a, 3-b, 1-
...read more.

Conclusion


Therefore, you now have a procedure for legally moving all n+1 disks to their new destination: follow the procedure I devised before, instead of moving the top n disks as a unit, I must use the legal amount of moves required to achieve the same effect. (In the example this will be 15+1+15)
In other words: if (x) is the minimum number of turns it takes to move n disks from one post to another, then  (x+1) = 2 (x) - 1.
I can prove that this is correct because the bottom disk will have to be moved sooner or later.


Conclusion
The formula a (x+1) = 2 a (x) + 1 seems fine, but the formula I need is one that only needs the number of towers (I have mentioned it above). I know that this formula is correct because 2a+1 (which is the least legal amount of moves that can be made to move 3 towers multiplied by 2+1) is 23-1.
\ 2n-1 must be the formula!

...read more.

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