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

Many students reported that the actual Josephus problem is more complicated than the one that we have been working on this week. There are two complications that we have ignored so far:

1. Every third person in the circle committed suicide, rather than every second person.
2. Josephus actually had a friend, and he and the friend wanted to be the last two people alive.
Let's try to study this problem. Assume that everyone is lined up in a circle, and starting with person number 3, every third person kills himself. We need to figure out the numbers of the last two people standing.

For example, if there are 3 people in the circle, then person number 3 kills himself, and the last two people standing are numbers 1 and 2. If there are 4 people in the circle, then first person number 3 kills himself, then person number 2. The last two people left standing are numbered 1 and 4.

1. As before, fill in the following chart:
   

   Number of people	Last two people left
   3	1, 2
   4	1, 4
   5
   6
   7
   8
   9
   10
   11
   12
   13
   14
   15
   16
   17
   18
   19
   20
   41

   
   
2. What patterns do you notice in this table? Can you explain any of these patterns?