Problem statement
In this problem, King Arthur was the ruler of Camelot and loved to play a game with his knights of the round table. He put numbers on the chairs around the table beginning with 1, and continuing with one chair for each knight. He had the nights sit down so that every chair was occupied, and he remained standing. King Arthur would then stand behind chair 1 and say "you're in". Next he moved to the knight in chair 2 and said "you're out" and that knight stood up and went to the side of the room. King Arthur then moved to chair 3 and told that knight "you're in". Next he went to chair 4 and said "you're out". The king went around the table in this manner. Once he got back around to chair 1, depending on if the previous knight was in or out, chair 1 would be in or out.
Process
o solve this problem, I started by creating a table with the values of the total number of seats and which seat won. It looked like this:
players winner
1 1
2 1
3 3
4 1
5 3
6 5
7 7
8 1
9 3
10 5
11 7
12 9
13 11
14 13
15 15
16 1
I immediately noticed a couple of patterns. The first one being that no even numbered seat would win this game, and the second being that all powers of 2, seat one will win. In the chart I highlighted the values where the numbers of players would result in seat 1 winning.
After talking with my group, we realized that we had to use a logarithmic equation to create a formula so that we can plug in any number of seats and find the seat that would win. Using a logarithm would give us the closest root 2 number, but we needed to find the rest of the equation to calculate the exact seat number that would win. After some turmoil, Mr B helped the whole class with the equation. We realized that it would end up being 2(x-2 ^[[log2^x]]) +1. To find the winner, plug in the number of players for x. Using PEMDAS the exact steps looked like, using 6 players as an example;
1)First, break it down to find the exponent first- [[log2(6)]]= 2.5 - The greatest integer function makes this answer 2, because that is the last whole number it passed on the number line. This becomes our exponent
2) Plug 6 into x, and 2 for the exponent - 2(6-2^2)+1
3) Solve this out, and you should get 5
players winner
1 1
2 1
3 3
4 1
5 3
6 5
7 7
8 1
9 3
10 5
11 7
12 9
13 11
14 13
15 15
16 1
I immediately noticed a couple of patterns. The first one being that no even numbered seat would win this game, and the second being that all powers of 2, seat one will win. In the chart I highlighted the values where the numbers of players would result in seat 1 winning.
After talking with my group, we realized that we had to use a logarithmic equation to create a formula so that we can plug in any number of seats and find the seat that would win. Using a logarithm would give us the closest root 2 number, but we needed to find the rest of the equation to calculate the exact seat number that would win. After some turmoil, Mr B helped the whole class with the equation. We realized that it would end up being 2(x-2 ^[[log2^x]]) +1. To find the winner, plug in the number of players for x. Using PEMDAS the exact steps looked like, using 6 players as an example;
1)First, break it down to find the exponent first- [[log2(6)]]= 2.5 - The greatest integer function makes this answer 2, because that is the last whole number it passed on the number line. This becomes our exponent
2) Plug 6 into x, and 2 for the exponent - 2(6-2^2)+1
3) Solve this out, and you should get 5
Solution
In this particular, problem there is no one solution, but one formula that solves for the number of seats you plug in. That formula being: 2(x-2 ^[[log2^x]]) +1
This formula works because, two to the logarithm and the greatest integer function, finds the closest root 2 number, where seat 1 will win, as proved in the chart. The 2 outside the parenthesis at the beginning accounts for the fact that you are counting by twos (in, out, in out) and the 1 at the end says that you started at seat 1.
This formula works because, two to the logarithm and the greatest integer function, finds the closest root 2 number, where seat 1 will win, as proved in the chart. The 2 outside the parenthesis at the beginning accounts for the fact that you are counting by twos (in, out, in out) and the 1 at the end says that you started at seat 1.
Evaluation/Reflection
In this problem, what challenged my thinking the most was starting the equation. I did not immediately understand why using a logarithmic equation would help find the solution, but I partly credit this to my limited knowledge of what a logarithm does before this problem. While working on this problem with my group, I realized that I needed to practice a bit on Khan Academy to help further my understanding of the elements of the equation. While working on logarithms, I realized its purpose in this problem! To find the closest root two number.
I also feel like I can recognize when I need to ask for help, and when I am available to help people. I don't think that asking for help is a weakness, or that you are not smart if you ask for help. The way I see it is, help yourself by helping others, and when you ask for help you are helping yourself.
I actually enjoyed taking the group quiz because my group was really supportive of people asking questions, so I felt like I could experiment with the problem without being judged. So that is exactly what I did, I was playing with the elements of the old equation, to try and see what fits with the group quiz's equation.
I would give myself an A on this unit because I think I was able to recognize where I needed help, and I used the resources I had (peers and Khan Academy) to ask questions and fit in more practice outside of school time. At the end of the unit, I feel like I really understand how to find an equation to this type of problem, and clearly explain what the elements mean in terms of solving the problem.
I also feel like I can recognize when I need to ask for help, and when I am available to help people. I don't think that asking for help is a weakness, or that you are not smart if you ask for help. The way I see it is, help yourself by helping others, and when you ask for help you are helping yourself.
I actually enjoyed taking the group quiz because my group was really supportive of people asking questions, so I felt like I could experiment with the problem without being judged. So that is exactly what I did, I was playing with the elements of the old equation, to try and see what fits with the group quiz's equation.
I would give myself an A on this unit because I think I was able to recognize where I needed help, and I used the resources I had (peers and Khan Academy) to ask questions and fit in more practice outside of school time. At the end of the unit, I feel like I really understand how to find an equation to this type of problem, and clearly explain what the elements mean in terms of solving the problem.