Problem Statement
King Arthur likes to play a game at a dinner table with knights. King Arthur first says to chair one "You're in". Then, he says to chair two "You're out". After he moved to chair three and said, "You're in". He said, "you're out" to chair four. He continued to go around the table in this manner. He went all the way around the table back to one and either said "You're in" or "You're out" to chair one depending on what he said to the other knight. He kept going around the table until there was one winner. Keep in mind that when the chair was empty, he skipped it.
Our Task
How would you quickly determine which chair to sit in so that you would win? The task is to find a pattern and formula.
Our Task
How would you quickly determine which chair to sit in so that you would win? The task is to find a pattern and formula.
Process
Individual Observations Individually, I started with underlining key points of the problem statement. You will see my main underlined points on the picture under "Things 2 keep in mind". Then, I asked one question to ask my group. I also made a t-table, and labeled the x and y. I did a mini illustration of four total knights. Group Observations With my group, we drew illustrations to further our thinking process. We also recorded the number of knights (x) to the winning seat (y) with the t-table. We noticed that the winning seat number is always odd. We also noticed that there are multiple 'restart' points, that means chair number 1 would win all over again when a certain number of knights would participate. For example... -With 2 knights, seat 1 would win -With 4 knights, seat 1 would win -With 8 knights, seat 1 would win again! -With 16 knights, seat 1 would win aggaaiiin! All these patterns would later contribute to the formula. |
Solution
In the process of finding the solution, we realized that there is exponential growth in every reset point. We also noticed that the reset points will always consist of a number of a power with the base of two. We also noticed the winning seat number is odd consecutive numbers until the reset point happens.
For example.. 1, 3, 5, 7, 9, 11, 13..etc until the next reset point Our class came up with this formula... General Rule:
You plug the total number of knights to x, subtract it from the nearest base of two number that is less than your number of knights, them multiply that from 2, and finally add one to the equation. (Total number of knights - the previous reset point of base two) =the difference between each other (the difference x two) +1= the winning seat |
Evaluation/Reflection
One of my group mates pushed my thinking by questioning it. For example, in the group quiz, I was trying to look for any patterns that could lead to the potential formula. Ways that I was challenged throughout the process was the interactions between two specific classmates and their lack of concentration. I would have to constantly explain the progress again because they weren't paying attention or contributing anything to the group work. Although, I am grateful I had to repeat what I said, this helped me reflect and come up with other observations and further my understanding of my findings. I asked my teacher Mr. Carter to give us a hint when our group was stuck. My group mate gave me confidence to ask for help. I, at times, find it hard to ask for help. I always think I can figure it out without anyone's help. Throughout the process, my group started off strong then would slowly decrease in productivity when we felt stuck or confused. When we asked for help, it brought us back on track and the cycle would repeat.
Overall, the group quiz affected my learning in a positive way. I forgot that I had so much bright minds around me to ask help from or learn from. I always felt I was alone on some kind of battle field with my math grade. This time, I felt secure and happy to say the least. I would grade myself a A-. I felt that I tried my best to engage the ones around me and question their thinking too. I didn't find a formula and for that I felt very disappointed with myself. I understood the concept and the main points of the problem. I just needed someone to guide me to finding the formula because I couldn't put the pieces together on my own.
Overall, the group quiz affected my learning in a positive way. I forgot that I had so much bright minds around me to ask help from or learn from. I always felt I was alone on some kind of battle field with my math grade. This time, I felt secure and happy to say the least. I would grade myself a A-. I felt that I tried my best to engage the ones around me and question their thinking too. I didn't find a formula and for that I felt very disappointed with myself. I understood the concept and the main points of the problem. I just needed someone to guide me to finding the formula because I couldn't put the pieces together on my own.