Around King Arthur's Table Problem
King Arthur would play a game with his knights, in which they would sit around a table and the king would decide whether the player was "in" or "out." He would start at the first chair and say "you're in" then the knight in the second chair would be "out" and so on. After going around the whole table once he would continue the pattern, determining the status of the knight in the first chair depending on what the chair before was. The last chair that was "in" would win a prize. The winner would change depending on how many knights were playing. Our task was to find a formula that could determine which would be the winning chair, given the number of knights playing.
Process & Solution
We were pretty confused at the beginning about how we could find a formula so we just began creating a table and finding winners by hand. We began to notice that there was somewhat of a pattern. First we found that every winning seat was an odd numbered seat because every even numbered seat was "out" in the first round. We realized that seat 1, won multiple times and had the same numbers after it but that the numbers after grew each time. We began writing our formula. We knew that our answers would have to be odd numbers so we multiplied by 2 and added one.
Our final formula is:
Knowing that the pattern reoccurred after "resetting" to 1, the beginning of our formula is about finding how many knights are in between the resetting point and the number of knights playing. Again, the last part of the formula is multiplying by the number of spaces between the resetting point and number of players by 2 and adding one to find the winning odd number that corresponds with the pattern. We know this works because using logarithms, we can find the winning seat for any number of players.
I've always been challenged in a different way when learning math through patterns and problems like this. I usually find myself getting impatient with observing the numbers and just trying things out. It is hard for me to stick with a problem where there isn't a clear correct answer from the beginning. I think what was really helpful that I did was ask other groups or listen to other groups ideas and then try to reverse engineer them to see if they were accurate or not. I think this makes a positive working environment where we are all helping each other. Something else I did when my group got stuck was that I pulled out a computer and graphed the coordinates in an online graph. The self advocacy I demonstrated put the problem in a visual context that was easier to understand for myself and my group. Although I was not present for the group quiz, I know that my group has different levels of mathematical abilities, but we are all active participants that contribute our multiple ideas, strategies, and perspectives. Again, when we get frustrated I know it is time to bring outside sources by asking other groups or Mr. Carter. I think that as a group and individually we have the ability to advocate for our needs. I would give myself an A for this problem because I was an active participant, made an effort to understand the formula and was able to ask for help when I needed it.