Towers of Hanoi
Introduction
============
We have been asked during this piece of coursework to investigate the
Towers of Hanoi. The Towers of Hanoi is a simple game whereby you must
move of a pile of 3, 4, 5 or any other number of discs (1, 2, 3, etc)
of decreasing radii from 1 of 3 poles to another pole (A, B, C).
You are only able to move one disc at a time and cannot place a larger
disc on top of a smaller disc. You must also complete this task in the
smallest amount of moves possible.
Our ultimate task was to complete the game with 4 discs and then 5
discs using the smallest amount of moves, then to find a formula to
find the smallest amount of moves for any number of discs.
Simple cases:
=============
4 discs
=======
After having tried to solve the puzzle with 4 discs I found that the
smallest amount of moves possible was 15. (See fig. 1)
5 discs
To try and make things slightly easier for myself I decided to use the
first 15 moves I had used for 4 discs and then proceed from there.
This method was effective and led me to find that the smallest number
of moves was 31. (See fig. 2)
Results and Formulas
====================
Number of discs
---------------
Number of moves
1
1
2
3
3
7
4
15
5
31
When placing all the results into a table I noticed that if you take a
certain number of moves for example 3 and then double it you end up
with 6. You only then need to add another 1 to make 7, which is the
next amount of moves. This works for any number of moves for finding
the next amount of moves. To simplify my findings I produced a formula
as shown below.
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...tries beside each other on the triangle can be added to get the entry below them.