Pt1420 Unit 7 Lab Report Conclusion

Conclusion

There were two main goals of this investigation. The first one was to find an equation that would produce the perfect number of moves for any number of disks that are being used in the puzzle. The second goal was to find a pattern between all of the different number of disk puzzles. The first goal of finding an equation was accomplished through trial, error, and logical thinking. By first graphing the data points, many equation types were able to be eliminated and a focus was put on exponential equations until the equation that worked perfectly with the data was found. The second goal was also accomplished in a similar manner. When there was not a clear correlation between the 3 disk puzzle and the 4 disk puzzle, the 5 disk puzzle

was tested and there was a more noticeable similarity.

The first 7 moves and the last 7 moves of the 5 disk puzzle are the same as solving the 3 disk puzzle. This means that the 7 disk puzzle will start with the same 31 moves for the 5 disk puzzle and the same 31 moves to finish the 7 disk puzzle. The 6 disk puzzle was also completed but there are not any pictures or commentary in this investigation. The 6 disk puzzle showed that the first 15 moves and the last 15 moves of the 6 disk puzzle are the same as solving the 4 disk puzzle. This shows that the odd number disk puzzles have the same pattern and the even number disk puzzles have the same pattern. This means that if the player knows how to solve the previous odd or even number puzzle, they can solve the beginning and end of the next odd or even number puzzle perfectly. Overall, the two goals are going to make solving this brain

puzzle easier.
Going back to the origin story, there were apparently 64 golden disks that when perfectly placed, the world would end. Using the equation, y= 264-1, this puzzle would take 18,446,744,073,709,551,615 moves. If one disk were to be moved per second, the puzzle would be completed in 5.85E11 years which is about 584 billion years. For comparison, the Earth itself has only been around for roughly 4.5E9 or 4.5 billion years.

Reflection
Based on the research and testing that was done in the exploration, a successful conclusion was reached.

The first goal of this internal assessment was to find an equation that would help to calculate the perfect number of moves for each puzzle no matter how many disks were being used. By testing how different function lines matched up with the three base points, there was a clear correlation and an equation was easily found. After the first goal was met, the second goal of this internal assessment was put into action. The second goal of identifying a pattern or algorithm to perfectly solve each puzzle every time was a more difficult challenge. Now that there is an equation for the number of moves and a recognizable pattern between the odd numbered disks and the even numbered disks, the goals of this internal assessment have been met. In the introduction, the story behind the Tower of Hanoi was explained. The story involved the 64 golden disks and the end of the world. If these disks existed, it would be interesting to see how moves it would take to be completed. When plugged into the function (y=264-1), the total number of moves was 18,446,744,073,709,551,615. Now that is an impressive number. This application was a fun twist on the original goals of this

investigation.
This topic interested me because I am a perfectionist and I always love to complete tasks flawlessly. Brain puzzles are always interesting and I have seen the Tower of Hanoi before but I had never completed the puzzles myself. This internal assessment was an opportunity to investigate something that fascinated me as well as allowed me to learn more about equations and patterns while having fun. When I was finding the equations, I think I really expanded my knowledge on exponential equations as well as my problem solving skills. I was able to come up with solution after solution when many of my original equations didn’t work. During this investigatigation, I also learned a good deal about patterns. The first couple of times that i was trying to solve the 5 disk puzzle, I hadn’t even realized that the first 7 moves were the same as the sequence to complete the three disk puzzle. Once I figured that out, the puzzle was easy to complete and it would make solving any other puzzle simple as well because if the puzzle can be started correctly, all of the moves leading up to the end solving sequence of the puzzle should have the same starting position of the previous odd or even puzzle. This means when doing the 7 disk puzzle, the first 31 moves will be the exact same as the 5 disk puzzle. The next 65 moves in the sequence will put the 5 smallest disks back on the first peg and then the last 31 moves will be the same as completing the 5 disk puzzle. All of these disk moving sequences will help to solve the 7 disk puzzle and these sequences will help to solve any larger odd number of disk puzzles. Overall, I think that the goals of this investigation were effectively met and that I learned a lot more that I thought I was going to and I think this IA was a success.