## 7th grade math: the nature of a circle

A problem, presented to me in 7^{th} grade:

100 people are in a circle. You remove every other person from the circle, leaving number 1, removing 2, leaving 3, removing 4 etc. You continue going around until only one person remains. Which will be the last remaining person?

The problem has a solution, and tedious ticking off will find it. There is, however a deeper answer, one that solves a circle with any number of people.

R=2(N-2^x)+1

R: the last person remaining

N: the total number of people in the circle

x: A value chosen where x is the highest value allowed such that 2^x is less than or equal to N.

eg if N=7, x=2, 2^x =4. If N=8, x=3, 2^x=8.

Therefore in a circle of 100 people, the last person will be number 73

In a circle of 10 people, the last person will be number 5

In a circle of 1000 people, the last person will be number 977

In a circle of 10,000 people, the last person will be 3817

I write this down now because I believe a more advanced mathematician should be able to use this equation as a starting point to find a deeper understanding of nature’s most perfect shape, the circle.

Please let me know if anyone develops anything.

Thanks,

-Evan