## Description of the Project & Our Task

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.

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.

## What made it most enjoyable problem of the year in math?

When introduced the problem and the details of it, I immediately found it interesting and intriguing! That was because the concept of how numbers or "knights", were taken out of a loop, and you had to find the last knight standing, was something that I knew was going to be a fun challenge. It was going to be a challenge because once you get into larger numbers, it gets extremely difficult to find the last knight without an equation that works for all numbers. You can see below what method you can do for smaller numbers, and you can find the equation for larger numbers on the actual page! It is under the Junior math section! I studied about this problem and tried coming up with equations some nights and I found an equation that worked! That moment I struck gold and it made me feel really good the next day where I presented my equation and how it worked. That is what it made it the most enjoyable problem for me this year.

## Easy Method for Small Numbers

You can go around the table counting and eliminating, who stays in and who gets eliminated.