*Mathematical Reasoning*, and my purpose is three-fold:

- Teach students how to write up a homework assignment in LaTeX
- Have students be comfortable writing correct proofs using set-theoretic notation: $\in, \bigcap, S=\{x\mid x > 0\}, \ldots$.
- Give students challenging problems that allow them space to be creative.

**Simplex Lock**

*How many different combinations are there on a Simplex Lock with 5 buttons?*

Students can generalize this problem to a lock with $n$ buttons. As far as I know, there is no closed formula for the number of combinations, $S_n$. Also, you can group combinations both by number of buttons used and by number of presses. Both of these groupings can lead to interesting recursive formulas. Students are used to getting explicit formulas, and this can be a refreshing departure.

**Mountain Road Traffic**

*On a certain mountain road, there are seven cars traveling, each at different speeds. Since the road is only one lane, there is no room to pass. Because of this, cars tend to clump behind cars that are traveling slower than they are. For example, if four cars were going, from rear to front, 50, 35, 45, and 40 mph they would eventually form two clumps: 50, 35 and 45, 40. The arrangement of clumps depends on the original order of the cars. By rearranging the cars' initial positions, how many essentially different clumps can the seven cars form?*

There are actually a number of questions one could ask students beside the one above:

- What is the average number of clumps?
- What is the average number of cars in a clump?
- How many distinct clump forms are there?
- etc.

Again, here there is no closed form that I've found, but there are certainly some patterns to investigate.