The Tower Of Hahnoi Analysis

Introduction

The tower of Hanoi is a brain puzzle that consists of three pegs and any number of different sized disks that fit onto all three pegs. This puzzle was created by a French mathematician, Édouard Lucas, but there is also a different origin story that uses 64 golden disks and relates to the end of the world. The legend says that when the 64 disk puzzle is completed perfectly, the world will come to an end. The goal of the game is to move all of the disks from the farthest left peg to the farthest right peg without placing a larger disk on top of a smaller disk. In the legend, the gold disks are the perfect weight so that if a larger disk were to be placed onto a smaller disk, the disk would shatter. There is an optimal amount of

The graph starts out close to the origin and the points seem to make a positive arc in the first quadrant. All of the points seem to follow a trend curve but the equation wasn’t clear yet. The first equation that was tested was y= arx, where ‘a’ and ‘r’ are constants. I plugged in the two points, (2,3) and (3,7). The final equation that these two points gave me was y= ⅓(3)x. When 2 is plugged in for x number of disks, the y equals 3 total moves. Although when 3 is plugged in for x, the y equals 27 which does not match up with the data so a new equation had to be tested. The second equation that was used was the equation of the trendline for the graph. The trendline equation is y= .706e.758x. When the first data point, (2,3) was plugged in, y was equal to 3.22. This number is not exactly 3 but it could be rounded down. The next point that was plugged in was (3,7). This data point was equal to 6.84 which was farther away from the original data but it could still be rounded up. The last data point that was tested using this equation was (4,15). In the second graph, it appears that this data point is the farthest away from the trendline while still being in contact with it. After plugging in the 4 for x, y equaled 14.64. Although all of these data points can be rounded up or down to their respective minimal moves, in reality, there cannot be .22 more of a move or .16 less of a move. The equation y=

The very first moves are identical to that of the three disk puzzle. This can be seen in the first of the five disk puzzle images. As the puzzle continues, by the seventh move, the three smallest disks have been moved to the farthest right peg. This means that to solve the five disk puzzle, you have first complete the three disk puzzle to move the last two large disks. This can be seen in the second image for this set. These moves seem to show that when there is an odd number of disks, the puzzle is solved by using the same pattern from the previous odd puzzle. For example, if this theory were true, the seven disk puzzle would be solved by first solving the three, then the five, and those two puzzles will help to solve the seven disk puzzle. Continuing on with the five disk puzzle, the next moves were not made by following a pattern because the only odd number puzzle that was completed before the five disk, was a three disk. The third image shows all of the disks except for the largest in the center. This action frees up the largest disk and it is then moved to the farthest right peg. This action is also directly related to the three disk puzzle. To complete the three disk puzzle, the two smallest disks had to be placed in the middle to move the largest disk of the three to the farthest right peg. After that action was made, the next step was to free up the second