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Introduction 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 Call know confirm that with four discs it is possible to get from the start (A) to the finish (B) or (C) in a minimum of 15 moves 2 4 6 8 10 11 12 13 14 15 I now know that it is possible to move a tower of 4 discs ina 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. 1 3 5 7 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 I now know that it is possible to move a tower of 5 discs in a minimum of 31 moves. I will know find the least amount of moves required to complete the game when you start with a tower of 2 discs 2 now know that it is possible to move a tower of 2 discs in a minimum of 3 moves. I will know find the least amount of moves required to complete the game when you start with a tower of 1 discs now know that it is possible to move a tower of 1 disc in a minimum of 1 move. I know have 5 results and will put together a table of results.
Table of Results 24 816 (32) As you can see there is no 2 4 8 constant pattern, but a diagonal 2 4 8 pattern. 2 4 2 From the top 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 to back up my prediction I can see that the previous term multiplied by 2 plus 1 gives the next e. g 7 x 2 + 1 = 15 15 x 2 + 1 = 31 31 x 2 + 1 = 63 I will know confirm that with six discs it is possible to get from the start to the finish in a minimum of 63 moves. 2 4 6 8 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 I now know that it is possible to move a tower of 6 discs in a minimum of 63 moves. So I can now construct a formula: The nth term = the previous term doubled, plus one or Tn = 2 Tn- 1 + 1 eg. 1 The 2 Tower 3 Tower 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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Research essay sample on 2 X X 2
