Problem Satement
The task is to find a general rule or pattern to determine which chair a knight should try to sit in to become the winner at King Arthur’s table. The way King Arthur chooses the winning knight is by sitting them at a round table and going around the table telling the first knight he is in and the next knight he is out and so on. The knights that are out are removed from the round table, and the rest stay in until King Arthur says, “You’re out.” Depending on what he says to the previous knight, the next knight is either in or out; therefore, if he says, “You’re out” to one knight, King Arthur will say, “You’re in” to the next knight around the table. King Arthur continues the game in this fashion, regardless of how many knights are at the table, until only one knight is left at the round table. The last knight remaining at the round table is the winner of the evening. There is a different winning “seat” depending on the number of knights there are, and in order to figure out which chair a knight should sit in to become the winner, one must find a pattern or formula. The task is to find this pattern.
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First Attempt
We made a chart with the number of knights across one axis, and seats marked by the king on the next axis. This let me see who would be selected if there were only one knight or if there were over 30 knights. A regular pattern emerged from that data, and I was able to put it into words.
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Process
I noticed 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 (21 =2 knights, 22 =4 knights, 23 =8 knights, 24 = 16 knights). The winning seats are odd consecutive numbers: 1, then 1, 3, then 1, 3, 5, 7, then 1, 3, 5, 7, 9, 11, 13, 15, and so on. When the number of knights is a power of 2, the pattern starts from 1 again; the pattern begins at 2, 4, 8, 16, and will again at 32. I also noticed the difference between the number of knights and the winning seat number was consistent
At the start of the pattern, the difference is always 1 less than the number of knights. For example, with 8 knights around the round table, the winning seat is 7 less than the number of knights: 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 9 – 6 = 3
The pattern can also be viewed backwards, starting with the last number, the number of knights before the next power of 2. For example, 15 is the number before 24 , or 16. Starting with 15, you subtract 0 to find the winning seat, and subtract 1 from 14 to find the winning seat to be 13, and so on.
At the start of the pattern, the difference is always 1 less than the number of knights. For example, with 8 knights around the round table, the winning seat is 7 less than the number of knights: 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 9 – 6 = 3
The pattern can also be viewed backwards, starting with the last number, the number of knights before the next power of 2. For example, 15 is the number before 24 , or 16. Starting with 15, you subtract 0 to find the winning seat, and subtract 1 from 14 to find the winning seat to be 13, and so on.
Solution
Even after finding various patterns in the math problem, it was challenging to find a set formula, but what I came up with is a strategy to determine the winning seat number regardless of how many knights are at the round table.
If you subtract from the number of knights the largest power of 2 less than it, multiply the answer by 2, and then add 1, you will arrive at the winning seat number. For example if there are 13 knights at the round table, and 8 is the largest power of 2 that is less than 13, you would do the following: 1) 13 – 8 = 5 2) 5 x 2 = 10 3) 10 + 1 = 11
The general rule you can follow to determine which is the seat a knight should try to sit in is by: 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
This general rule can be made simpler and better if there were a clearer algorithm or formula one can follow instead of working through these many steps, but with greater knowledge in mathematics, one can find a more efficient formula.
If you subtract from the number of knights the largest power of 2 less than it, multiply the answer by 2, and then add 1, you will arrive at the winning seat number. For example if there are 13 knights at the round table, and 8 is the largest power of 2 that is less than 13, you would do the following: 1) 13 – 8 = 5 2) 5 x 2 = 10 3) 10 + 1 = 11
The general rule you can follow to determine which is the seat a knight should try to sit in is by: 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
This general rule can be made simpler and better if there were a clearer algorithm or formula one can follow instead of working through these many steps, but with greater knowledge in mathematics, one can find a more efficient formula.
Reflection
In this problem at first I thought it will take me less than 10 minutes to solve but then after trying and trying I felt challenged and like this the problem pushed my thinking in order to find a certain pattern, I will admit I had a really bad habit and that was not trusting my peers because I often think I know way more than what they do and I dont open myself towards them trying to help me, I never felt confident to ask because of me looking like I dont know but also because I was pretty sure they will not know either, but I noticed many things that helped us get to the solution and I felt really comfortable o show my peers my progress, in the group quiz I found out most of the things, it did a change in my learning in a good way because knowing it was a test made me put ay more effort on the situation/problem, if I had o grade myself I will give me a 8/10.