Problem Statement
Our Task:
First, King Arthur put numbers on the chairs beginning with 1 and continuing around the table, with one chair for each knight. He had the knights sit down so that every chair was occupied. King Arthur himself remained standing. Then King Arthur stood behind the knight in chair 1 and said,“You’re In.”Next, he moved to the knight in chair 2 and said,“You’re Out,”and that knight left his seat and went off to stand at the side of the room to watch the rest of the game. Next, he moved to the knight in chair 3 and said,“You’re In.”Then he said,“You’re Out”to the knight in chair 4, and that knight left his seat and went to the side of the room. The king continued around the table in this manner. When he came back around to the knight in chair 1, he said either “You’re In”or “You’re Out,” depending on what he had said to the previous knight. (If the previous knight was “In,”then the knight in chair 1 was now “Out,”and vice versa.)
The king kept moving around and around the table alternately saying, "You're In" or "You're Out" to the knights who remained at the table. (If a chair was now empty, he simply skipped it.) He continued until only one knight was left sitting at the table. That knight was the winner.
Of course, the number of knights varied from day to day, depending on who was sick, who was chasing dragons, and so on. Sometimes there were only a handful, and sometimes there were over a hundred!
Your task is to develop a general rule, formula, or procedure that will predict the winning seat in terms of the number of knights present. Be sure to explain why your rule works.
ProcessIndividually
When tackling this problem on my own, I immediately drew my own simulation of knights around a round table. Starting at chair one, I simply decide who stays in and who gets eliminated. If we are starting with one as "in", and we alternate from there, I figured out that all evens will be eliminated in the first round. Then I created a table, stopping at sixteen. I created a table trying to see if there was a pattern that I could make a formula out of. I noticed that there was a reset point which would hint at me what needed to be included in my formula, 2^! Later that day I ended up really analyzing that table and I started trying to come up with formulas and I found it! As you can see the formula, I came up with was number of knights subtracting the closest power of 2 then multiplying it by 2 and then adding 1.The solution will also be listed below in the "Solutions" section.
ProcessGroup
After solving that problem individually, we tried a different variation of the same problem with a group to grow collaborating skills, and see if we can come up to a solution together. We were given the same problem, but flipped. The first seat chosen would be declared "out" and the next seat after would be considered "in". Because it was essentially the same problem, we started off the same way.
As a group we know that this problem was very similar to the last one except we are starting by telling chair number one that they are "out". We tried many numbers of knights to find patterns. This told us all odds would be eliminated in the first round. We also found the formula to be very similar to the last problem, but faced some troubles with perfect squares such as 4, 8, and 16. I contributed lots of work into this group problem as well as the individual, and it showed because my group was making lots of progress and we were asked by our math teacher to show that progress to the class.

Solutions
Individual:
The solution to the individual would be written out like this W=(K2^)x2+1. To say it out loud it would be said like this, "Number of knights subtracting the closest power of 2 then multiplying it by 2 and then adding 1. This would work for any umber of knights to get you the winner of that round.
Group:
To clarify again, the first seat chosen would be declared "out" and the next seat after would be considered "in". We found our winners, then found our reset. Because the winners were all even numbers, the reset would occur when z was squared. For most of the time given to find the problem, we spent trying to remember what our last equation was. Once we remembered "2 (x  2 [[log^2x]])" we applied it to our table and took out the "+ 1". When plugging in, we found a hole within our solution. It didn't apply to the reset. When we finally came together as a class, we were given the solution that would work for any value 2 (x  2 [[log^2(x1)]]).
The solution to the individual would be written out like this W=(K2^)x2+1. To say it out loud it would be said like this, "Number of knights subtracting the closest power of 2 then multiplying it by 2 and then adding 1. This would work for any umber of knights to get you the winner of that round.
Group:
To clarify again, the first seat chosen would be declared "out" and the next seat after would be considered "in". We found our winners, then found our reset. Because the winners were all even numbers, the reset would occur when z was squared. For most of the time given to find the problem, we spent trying to remember what our last equation was. Once we remembered "2 (x  2 [[log^2x]])" we applied it to our table and took out the "+ 1". When plugging in, we found a hole within our solution. It didn't apply to the reset. When we finally came together as a class, we were given the solution that would work for any value 2 (x  2 [[log^2(x1)]]).
Evaluation/Reflection
I understood the main pattern immediately, but I struggled with translating the problem into an equation. Without knowing how "log" could be implemented, there was no way I could have come to this conclusion alone. I had great group members like Aaron that reminded me that we learned about "log" not to long ago. Having a group to help make sense of things was also useful as I was pushed to explain my ideas to the people around me. If I were to grade myself on this assignment, I would give myself an A+. Throughout the couple days working on this problem, I was able to understand the content, as well as explain it to my group to further comprehend the work I was doing.