The Towers of Hanoi.

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The Towers of Hanoi

The Towers of Hanoi is a mathematical game.  It consists of 3 poles in alphabetical order and 3-7 discs numbered in size order.  At the beginning of the game all the discs are on pole A in size order.  The object of the game is to transfer the entire amount of discs from pole A to pole B or pole C in the minimum amount of moves possible.

These are the rules of the towers of Hanoi:

  • A disc must never be placed on top of a smaller disc than its self.
  • Only one disc may move at a time.
  • The discs must be on a pole at all times except for one moving

By looking at the table it is clear there is a pattern.  

Every move that is made doubles the last number of moves and adds one.

The algebra equation: 2x+1.  (Where x is the previous amount of moves)

The tower with 3 discs makes 7 moves.

Therefore using the rule 2x+1     x = 7

2*7 + 1= 15 = no. moves made in a 4 disc tower.

Each time a tower is built it follows the same pattern as the one before as it uses the same moves to produce a Hanoi tower until the additional disc is moved.  Then the additional disc moves once and the rest of the discs repeat, building the tower but on the other pole to form the Hanoi tower from that of which the final Hanoi tower is built.  

The tower is built the same as the previous tower in the sequence.  Although after the tower is built the new disc is moved to pole B or C.  This move is where the plus one comes from the equation as it is a separate move that is not a part of the unbuilding or rebuilding process of the previous tower.  As this disc is an extra, which was not in the previous tower, this disc’s move is the change which allows the tower to be rebuilt on the opposite pole.

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Example

For example in 4 discs:

Pole A                Pole B                Pole C

4321

432                                                1

43                        2                        1

43                        21                        

4                        21                        3

41                        2                        3

  1. 32
  1. 321

Begins to change as the additional disc is used and an extra tower has to be rebuilt and un-built.

  1. 321
  1. 32

2                        41                        3

21                        4                        3

  1. 43

2                        43                        1

  1. 1

4321

This pattern occurs because in order for a 4 disc tower to be built a 3-disc tower must be built then un-built.  The added 1

The 1st disc moves to pole C and creates a tower the same to the 3-disc tower.  When ...

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