KING ARTHUR'S ROUND TABLE

Problem Statement:

Every night King Arthur invited a group of knights to sit at his round table and to win a prize. Whoever sat in the right chair at the table would win and the winner’s chair changed depending on how many knights there were. He would go to the first chair and say “You’re in.” Then, he’d go to the second chair and say “You’re out.” He would say to the next chair “You’re in,” then after that “You’re out,” to the next knight, and so on and so forth. What we had to do was come up with a general rule that would entail a win every time.

Process:

First, I tried to see which chair would win for a couple different numbers of knights. I tried two through ten and created a table to see if I could find a pattern. The x represented the number of knights at the table and y represented the chair number that would win. At first, I felt like there was not a pattern because I was just looking for the numbers on the y side of the table to increase, however, I didn’t see that in the table. Next, my table group came together to share our thoughts.

We first all came to the conclusion that the winner would never be an even number because King Arthur started at the chair number one and then took out two and then kept in three. This “in-out” pattern meant the winner would always be an odd number. We decided to put the data from the table onto a graph and we noticed that the graph would always rise and fall back to the chair winner, one. We found that for each power of two, the winning chair would be one. Then we realized that it was a kind of “reset” number which was where we would count off by two from the power of two that it just passed.

Solution:

We first wrote the solution out as 2(K-2^x)+1. Where 2 was how many times you move up by two, k was the number of knights, 2^x was the power of 2 it had just passed, and 1 made the solution an odd number.

We later were taught that it was also written as 2(x-2 [{ log2^]}+1).

Reflection:

When starting this problem I felt that I knew what to do to try to look for a pattern and then when we discussed it as a group, we found one. I felt extremely confident that I could figure out what seat would win given the number of knights, but I felt really stuck on how to find an equation that would work for all numbers. As I would talk to my group mates, I would find small things, but ultimately I felt like we were going in circles when we would end in the same place of just counting up or down by twos. During these moments, I felt extremely challenged mathematically. I feel like I am able to recognize when I struggle and I am confident enough to ask for help. I am confident in my strengths and understand that I may not be the only one who has the question I want to ask. Along with this, I feel extremely comfortable in the environment we have created as a class which is one of just wanting each other to get better at math. During the final stages of the problem, I felt my group worked together very well. We were all able to help each other fully understand and asked clarifying questions to each other to make sure no one was left behind. The group quiz had a positive effect in my learning because it really challenged me and my whole group to take what we knew from the last problem and apply it to the new one. Although we didn’t come up with an equation, we all worked together to create general rules and in the end we were all anxious to know the answer because we were extremely focused and driven to know what we were missing. I would give myself and A on this problem because I feel that throughout the entire process I was working well with my group and pushed myself to continue looking for patterns even when I was a bit frustrated with the results I was getting.

​