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Warm Up Problems for Raw Recruits

1. When the sergeant gives the recruits the "Left Face" order, it is possible for the recruits to choose directions such that they all think that they are correct and none of them turn around after one second. Find an example of this situation. How many of these initial positions are there? Describe them.

2. You walk by and see the following arrangement of recruits:

If this is not how they were arranged at first, what arrangements might have existed a second earlier? How many seconds ago might they have begun? Try answering these questions for other arrangements. Are there positions that can only be starting positions (i.e., that will never appear a second after the "Face Left" order)?

3. If each of the six recruits chooses to face left or right at random, how many possible first positions are there? How many starting positions are there with *n* recruits?

We do not include results for these warm up problems on the Making Mathematics web site.