King Arthur had a round table at which he and his knights would gather. On special nights, when the kind had a special task for one of the knights, it was very hard for him to understand which knight to give it to. The king would use a simple pattern to determine who would be awarded with the task. Each chair was numbered and the king would start at number 1, count them in, go to chair number 2, count them out, then to chair 3, count them in and continue this pattern until there was one knight left. The pattern was simple, but it was hard to determine which seat would win because some of the time there would be a couple knights and on other nights there would be over a hundred.
Our Task
We were assigned to create a rule for this pattern to know which seat would be the winning seat no matter how many knights there were.
Initial Work
My first instinct with this problem was to physically draw out the scenario that was described. I made a round table that included a random number of knights, and went in a circle according the pattern. This was essentially what my group and I did until we got to 27 knights, because we realized it was going to take too long. By this time we had figured out a couple rules that we could knew we could rely on. One of them being that the answer would never be an even number, always odd. Another rule we found was that when the winning chair was one, it would be an exponential of 2. For example 2^2, 2^3, and so on.
We also discovered a pattern within the number of the winning seats. The seats would go up every odd number until the pattern "reset". These reset points were at the other rule we found. Every time there was an exponential of 2, the winning seat would be one and the pattern will continue from there.