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

1. Take the list of numbers answering the question about the Josephus problem from the last homework. Rewrite both columns of your chart in base 2. Do you see a pattern?
2. In class, we tried to generalize the concepts of EVEN and ODD to 3EVEN and 3ODD. In this problem, we will try to see if we can extend those ideas any further.

   Define a number as 4EVEN if it is a multiple of 4. Call a number 4ODD if it is not 4EVEN.

   What parts of the following charts can you fill in? Which parts are ambiguous? Be sure to justify all of your answers, using the same sort of reasoning as we did in class for EVEN and ODD.
   

   +	4EVEN	4ODD
   4EVEN
   4ODD

   
   

   *	4EVEN	4ODD
   4EVEN
   4ODD