**Initially this exercise is to 'loosen up' your brain, free it from the
shackles of the real numbers, and let you begin to explore other
structures. Eventually, you will repeat this exercise (with perhaps some
modifications) as part of the midterm and final exams.**

**Consider the following:**

**1. A square piece of paper folded in in half horizontally,
vertically, and on the diagonals, with a little star on it as shown below:**

** Variation: What if you have another regular
polygon, such as an octagon or an equilateral triangle?**

**2. A clock with 12 hours and 60 minutes per hour.**

** Variations: What about other amounts of hours or
minutes per hour?**

**3. A two-bit string, ie (_, _), where the _'s can be
either a 0 or a 1. 0+0 = 0, 0 + 1 = 1+ 0 =1, and 1+1=0, and you add
component-wise, so that e.g. (1,0) + (1,1) = (0, 1).**

** Variations: What if the string is longer, eg (_,
_, _) or (_, _, _, _)? What if you have more than 0 and 1? Say
you have 0, 1, 2. What should 1+2 =? What about 2+2, etc?**

**Write up as much as you can possibly discover about these three objects and
their variations. Think carefully about your choice of notation as you
write down your ideas.**

**Due: Monday, September 8.**

**Note: The background for this page is from http://xaravve.trentu.ca/330/pictures.html#pic.cayley.
You will understand it (it is very cool) by the end of this course!**