## Problem Statement

In this problem, we were tasked to find a rule or pattern to find which chair a knight should sit in to be the winner of King Arthur's table. King Arthur chooses the winning seat by sitting knights around a round table, then he goes around the table telling the knights if he they are in or if they are out. He continues this until there is only one knight left, and that is the winning knight. In order to find out which chair a knight should sit in to be the winner of the game, a pattern or a formula had to be found. We were tasked with find it.

## Process

I began this problem by making a list of the number of knights to find a pattern. After listing the first 16 winning seats, I was able to find a pattern. I was able to notice that the winning seat is always an odd number, and that there is a pattern at the start of each power of 2 number of knights.

At the start of the pattern, the difference is always 1 less than the number of knights. For example, with 8 knights, the winning seat is going to be 7 less than the number of knights. It would look like 8 - 7 = 1. The next winning seat has a difference of 2 less than the number of knights at the start of the pattern:

At the start of the pattern: 8 knights 8–2=6

The next number of knights: 9 knights 9–6=3

At the start of the pattern, the difference is always 1 less than the number of knights. For example, with 8 knights, the winning seat is going to be 7 less than the number of knights. It would look like 8 - 7 = 1. The next winning seat has a difference of 2 less than the number of knights at the start of the pattern:

At the start of the pattern: 8 knights 8–2=6

The next number of knights: 9 knights 9–6=3

## Solution

After finding multiple patterns, I was still stuck trying to find a formula. What I came up with is a way to find the winning seat number regardless of how many knights. If you subtract from the number of knights from the largest power of 2 less than it, multiply the answer by 2, and then add 1, you will then find the winning seat number. For example if there are 13 knights and 8 is the largest power of 2 less than 13, these would be your steps:

1) 13–8=5

2) 5x2=10

3) 10+1=11

The winning seat would be 11 if there were 13 knights.

The general rule you can follow to find which is the seat a knight should sit in is by following these steps:

1) find the greatest power of 2 that is less then the number of knights

2) subtract that number from the number of knights

3) multiply that number by 2

4) add 1 to the answer

## Evaluation

This problem was one of the ones where I actually did manage to push myself more to think in it. I was constantly asking my group members questions. Before we had our group quiz on this, I was already making our group come together in the problem. I worked with everyone in my group to try to see what they were doing and how we could all bring each of our ideas together which we eventually did. When the group quiz came along, I was asking more questions that ever in my group. I even got credited because I asked one of my group members "Why did you go "Ooo"". I had no shame in asking any questions through out this whole problem. If I needed help or was stuck, I would try to get it taken care of right away to keep moving forward faster. From getting all this help, I feel like I am able to even help other with this problem. My group worked very well together and we were able to all confidently work.

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