Problem Statement
king_arthur_problem.pdf | |
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Process
When we were first introduced to this problem we were given 5 minutes to brainstorm on ways to figure out how to solve this problem mathematically or write down any questions we had. My first attempt was to try to write the rules and order King Arthur would do. I tried drawing a round table to figure out how the "game" would work out. Lastly I wrote down some questions on things about the problem that were unclear to me.
When we were first introduced to this problem we were given 5 minutes to brainstorm on ways to figure out how to solve this problem mathematically or write down any questions we had. My first attempt was to try to write the rules and order King Arthur would do. I tried drawing a round table to figure out how the "game" would work out. Lastly I wrote down some questions on things about the problem that were unclear to me.
After that we shared our attempts at trying to solve the problem with our table. We all tried very similar methods, one of them being writing each and every winner from seat 1-16. We we noticed some patterns but we could not pin point on how they connected, one of the patterns we found were that no matter what the winning seat was going to be odd. Eventually one of my groupmates mentioned how there was a pattern of the winning seat and the number of the knights. When 2 knights played, knight #1 won, when 3 nights played, knight #3 won. We noticed that it would reset back to one. Eventually we figured out that it adds by 2 each time and that it resets each time that a number is to the power of 2. For example, when there was 2 knights the winner was 1, when there was 3 knights it reset to one, when there was 8 knights it reset to one again. We came to the conclusion that 2x2=4 and when there was 4 knights it reset to 1, 4x2=8, when there was 8 knights it reset to 1 so we figured out the pattern all we had left to do was find an formula that would help find the winning seat faster.
Solution
We knew the pattern so we used a table to try to see how long it would take to reset after the numbers got bigger . We did a table all the way until we reached the number 7o, by then it had reset 6 times. Eventually tried different formulas and one worked out. x
w=(k-2 ) 2+1
w stands for the amount of winning seat, k stands for the amount of knights, the 2 x stands for the closet place it resets at. For the x for 5 would be 2 because it resets to one at 4 and it's the closest to 5. We tried it out with a small number like 5.
2
w=(5- 2 ) 2+1 which simplified would be w=(1)3 which equals 5 and is the correct answer.
At first we were able to solve the problem with small numbers, but then we realized that it would be hard trying to solve it with big number because you wouldn't know what the closest number it reset at would be. Eventually Mr.Carter taught us how to find where the closest number it resets at could be by using log 2. So if we tried to figure out what the winning seat for 70 knights would be, we first would type in log 2 70. Which equals 6 and that's x
x
w=(70-2 ) 2+1 simplified would be w=(6)3 which equals 18
We knew the pattern so we used a table to try to see how long it would take to reset after the numbers got bigger . We did a table all the way until we reached the number 7o, by then it had reset 6 times. Eventually tried different formulas and one worked out. x
w=(k-2 ) 2+1
w stands for the amount of winning seat, k stands for the amount of knights, the 2 x stands for the closet place it resets at. For the x for 5 would be 2 because it resets to one at 4 and it's the closest to 5. We tried it out with a small number like 5.
2
w=(5- 2 ) 2+1 which simplified would be w=(1)3 which equals 5 and is the correct answer.
At first we were able to solve the problem with small numbers, but then we realized that it would be hard trying to solve it with big number because you wouldn't know what the closest number it reset at would be. Eventually Mr.Carter taught us how to find where the closest number it resets at could be by using log 2. So if we tried to figure out what the winning seat for 70 knights would be, we first would type in log 2 70. Which equals 6 and that's x
x
w=(70-2 ) 2+1 simplified would be w=(6)3 which equals 18
This problem challenged my thinking differently than all the other problem we've done. It challenged e differently because I knew what the problem was asking I just had a really hard time trying to figure out how to make a formula. I also felt challenged sometimes because at times I was afraid to ask my group mates for help, since I din't know them very well. Which is something that I need to work on, talking to my group mates and asking more questions even if not that close to my table mates . Although I feel like at times I definitely was being an active listener and sometimes would share my progress and try to connect the connections. Since this problem challenged me with trying to come p with a formula I was able to help one of my table mates with the problem. Overall I think I would give myself a B+ on this problem because I was on task most of the time trying to figure problems ,but I know that I could've been actively speaking and trying to collaborate more that I did.