MT 007.02 / SL 266.01
Ideas in Mathematics: The Grammar of Numbers

1. Write a short paragraph about Josephus. Be sure to list your sources.

   Remember the definition of the Josephus problem: a group of men gather in a circle, and every alternate person kills himself, starting with person number 2. Our job is to figure out the last person left standing.

   If there are n people in the circle, then we use the notation J(n) to represent the last person left standing. For example, if there are only 2 people, then person number 2 kills himself, and person number 1 is left. Our short-hand representation of this fact is to write J(2)=1.

   To take another example, if there are 3 people in the circle, then first person number 2 kills himself, then person number 1, so person number 3 is the last person left alive. Our concise representation of this is to write J(3)=3.

2. Complete the following chart:
   

   n	J(n)
   2	1
   3	3
   4
   5
   6
   7
   8
   9
   10
   11
   12
   13
   14
   15
   16
   17
   18
   19
   20
   41

   
   
3. What patterns can you notice in these numbers?