Investigating Towers of Hanoi Mathematics coursework

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Investigating Towers of Hanoi
Mathematics coursework

Surname: Thomas
First name: Jawhari


Candidate no: 7160
Centre no: 14368


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. The left most pile is the starting pile you need to move the discs to the right most pile, never putting a disk on top of one, which is smaller. The middle pile is there for intermediate storage. The piles work like a triangle and a disc can be moved from one pile to any other pile.


Looking for the Pattern. 

To figure out how many turns it will take for more than four disks, and to figure out how long it will take the monks to finish their task, you need to find a pattern relating to the number of disks to the minimum number of turns it takes to win the game. Firstly I will have to find the minimum number of moves it takes for me to finish the game with 1-7 discs.


Justifying the amount of moves.

I have compiled an archive that show the size of a pile and the minimum amount of move that it will take to the pile this archive is listed below. I am using this archive to prove that I completed each game using the least amount of moves possible:

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  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, ...

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