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# Towers of Hanoi

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Introduction

## Towers of Hanoi

I am going to investigate the Hanoi Towers. The objective of the game is to move the discs from position A to positions B or C in the minimum number of moves where in one go you are only allowed to move one disc. I will vary the number of discs and record my results. I will make predictions and look for patterns. I will ultimately aim to find a formula for the number of moves it takes to move the tower from A to B or C.

Suppose you just had one disk. Then the method for moving that one disk from the starting spindle to the ending spindle is simple: just move it. It takes one step :

Now suppose you had two disks.

Middle

I now know that it is possible to move a tower of 4 discs in a minimum of 15 moves. I will know find the least amount of moves required to complete the game when you start with a tower of 5 discs.
I now know that it is possible to move a tower of 5 discs in a minimum of 31 moves.

Conclusion

s

1

3

7

15

31

2     4    8     16

From the bottom row of differences I can see that the previous term doubled gives the next term in the sequence. Therefore I predict that the next difference would be 32 for a tower of 6 discs (see results table) giving the number of moves as 63 also I can see that the previous number of moves multiplied by 2 plus 1 gives the next number of moves e.g.

1 x 2 + 1 = 3
3 x 2 + 1 = 7
7 x 2 + 1 = 15
15 x 2 + 1 = 31
31 x 2 + 1 = 63

So I can now construct a formula:

The nth term = the previous term doubled, plus one

or

Tn = 2Tn-1 + 1

eg. 1 Tower 2 x 0 +1 = 1
2 Towers 2 x 1 +1 = 3
3 Towers 2 x 3 +1 = 7
4 Towers 2 x 7 +1 = 15
5 Towers 2 x 15 +1 = 31
6 Towers 2 x 31 +1 = 63
7 Towers 2 x 63 +1 = 127
8 Towers 2 x 127 +1 = 255
9 Towers 2 x 255 +1 = 511
10 Towers 2 x 511 +1 = 1023

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