Problem Statement
In this problem King Arthur would invite all of his knights to parties around his table, at these parties they would play a game. The game went like this, King Arthur had numbered each chair around the table starting with one and continuing forward. There was a chair for each knight. King Arthur stayed standing and all the knights occupied a chair. He went behind the knight in chair one and said "you're in" then moved over to the knight in chair two and said "you're out" he continue this pattern all around his table. When a knight was out they had to get out of their seats and stand off to the side, if a chair was now empty he would just skip over it. He continued until there was only one knight at the table, that knight would be the winner.
Our task was, if we knew the amount of knights there was going to be at the table , how could we figure out which chair would be the wining seat.
Our task was, if we knew the amount of knights there was going to be at the table , how could we figure out which chair would be the wining seat.
Process
At first we were given a few minutes to ourselves to investigate the problem and see what we could figure out on our own.The first thing I thought to do was to check who would win in a table with 2 seats and then continued going up until I started to notice a pattern. I noticed almost right off the bat that the even numbers would be eliminated first, which meant an odd number was always the winner. After that we talked with our groups, together we were able to figure out that there was a pattern in the answers. It would go 1,3,5,7,9,11,13,15 and that every time the number of knights matched the number of the winning chair, the following winning chair would be one and the pattern would restart. We also noticed that the difference between each of the times it would start over was increasing by powers of 2. For example from 1,1 to 3,3, it was 2 and from 3,3 to 7,7 it was 4. We took an educated guess and assumed that the next time it would start over it would go up to 8, and it did and the next time it repeated was right after 15,15.
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Solution
The rule we came up with in class was the winning seat would be the closest power of two subtracted by the number of knights playing multiplied by two and then a one added at the end. For example if we wanted to find out who would win with 23 knights playing you would do 2(23-16)+1. Because of pemdas first we would solve what was in the parenthesis, 23 -16 (16 because it's the closest power of two to 23) would equal 7. Then we would multiply the seven by two which would give us 14, finally we would add the 1 that was at the end and it would give us fifteen. This tells us that with 23 knights present, you should sit at chair number 15 to win.
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Reflection
Word problems like this always push my thinking because I have to think of a way to solve the problem based on all of the information given to us. I feel like I often let myself get frustrated, especially in word problems, when I'm having a difficult time finding patterns and solutions. In this situation I don't think I was challenged too much in group work but I do feel like I had a hard time getting started on the problem because I was scared of doing something wrong or not finding the right pattern. Once I got over that, I was able to identify patterns and work with my group mates to expand on what we found. I always know that if I need help I could ask my group mates and they usually will be willing to help me understand the problem we are working on. I feel completely comfortable asking for help because I want to understand what we do in class and I find it exciting when I finally understand a topic I have been struggling to understand. I love helping others when I feel comfortable in the concept because I get to share what I know and my way of thinking with my pals. Even if I don't understand the topic I will be willing to struggle through a problem with one of my peers so that we could build off of each other and eventually understand the problem together. The group quiz was really helpful because I felt like I was able to see how well I personally understood the concept and how well my group understood it by taking the same problem we had worked on but changing the "rule". In the group quiz it was so cool to see us all work together to reach a deeper understanding of the problem. I would give myself an A+ because I tried my best in everything I did and kept moving forward even when I made mistakes. I also worked very well with my peers and grew not only as a math student, but also as a group mate. I had a lot of fun during this problem!