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?