King Arthur has a large amount of knights, each night a different number. He had a game he would like to play with them to figure out which knight would have to chase the next dragon, etc. The game would start at chair one. The king would tell person one that they were "in" and the next that they were out. He would continue this pattern around and around the table until there was only one knight left, the winner. Our task was to find out a way to determine who would win just by knowing how many knights there were to start with.
Process
As a group, we organized our data into X and Y tables and solved through drawing dots and repeating King Arthur's process. At first, the pattern seemed random, but we then saw that it would follow a pattern of going through each odd number, starting back at one every time there was a root 2 number of people. I wasn't able to see this on my own though, it was only when we shared as a group that I caught on. From that point, we tried to make a formula that replicated the process we now could do in our heads: Find the nearest root two number below your starting number and subtract it. Then, count up by odd numbers however many spaces were between the two.
Solution
One of our biggest struggles was finding the nearest root 2 number, but eventually we got to this function: [[log2X]]. This uses the greatest integer function which basically takes the decimals away from any number. For example, log223 is 4.something, but we don't want the exponent that will get us to 23, we want whatever the nearest root 2 number is. So [[log223]] is 4, and 2^4 is 16. Then we had to solve the next part. We know where the pattern started at one for this cycle of counting by odd numbers, so we then subtract our 2^[[log2X]] from x. This gets us the amount of times we must count up by two, starting at 1. So counting by two mathematically is multiplying by two and starting at 1 is adding 1.
Answer: 2(x-2^[[log2X]])+1
Evaluation/Reflection
My favorite part of this problem was during the group quiz. I really enjoyed working with my group and bouncing ideas off of each other. It was helpful having other minds around me that were able to recognize patterns I was unable to see on my own. I was definitely able to ask for help, help others, etc. because our group was always interacting with each other. Samantha and I especially were really building off of what the other was saying and we eventually came to a pretty close solution. The problem was slightly different for the quiz. It was if King Arthur started on out instead of in, so that 2 and all even numbers stayed in instead of the odds. In the end, we got 100% with one equation that worked for everything except root 2 numbers which all equaled themselves, but I was still curious because Mr. B told us that there was one solution that would solve for all possibilities. This was when I felt inspired. I actually sat down and kept working because I just genuinely wanted to solve the problem. I liked how it pushed my thinking from, "how can I find a formula" to "how can I take what I know about the process of solving this problem and turn that into a formula." In the end, this way of thinking is what helped me solve the problem fully.
I would give myself an A+ on this problem because even when we knew we had done what we needed to do to get a good grade, I still continued to challenge myself. I leave this problem with much more confidence in my ability to gather information, recognize patterns, and form equations based on my collected data.