Here's something I found a while back just messing around. Since it relates to the course, I felt I'd make some puzzles and not let John have all the fun.

Consider the following function on the integers,

$$\chi(m,n):= \left \lceil \frac{1}{m-n+\frac{1}{2}} \right \rceil -\mathrm{sign}\left ( m-n+\frac{1}{2} \right )\cdot \left \lfloor \frac{1}{ \left | m-n+\frac{1}{2} \right |} \right \rfloor.$$

**Puzzle 1:**: What are some interesting features about the function
\$$\chi(n,m) \$$?
Does this function have a common name?

Now consider the following function on the integers,

$$E(m,n) := \chi(n,m)+\chi(m,n)-\chi(n,m)*\chi(m,n).$$

**Puzzle 2:**: What are some interesting features about the function
\$$E(n,m) \$$?
Does this function have a common name?

Now consider the following two functions on the integers,

$$\mu(m,n) := n * \chi(m,n)+m * (1-\chi(m,n)),$$

and,

$$\mu'(m,n) := m * \chi(m,n)+n * (1-\chi(m,n)).$$

**Puzzle3:**: What do the functions \$$\mu(m,n) \$$ and \$$\mu'(m,n) \$$ do?
Do these functions have common names?