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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?

## Comments

The first thing I notice is that \(\chi(m,n)\) can be replaced by a function of a single variable \(\psi(m-n)\). This also means that flipping the arguments to \(\chi\) just flips the sign of the single argument to \(\psi\). Is there a reason you are using the two-argument version here instead?

`The first thing I notice is that \\(\chi(m,n)\\) can be replaced by a function of a single variable \\(\psi(m-n)\\). This also means that flipping the arguments to \\(\chi\\) just flips the sign of the single argument to \\(\psi\\). Is there a reason you are using the two-argument version here instead?`

There is no particular reason. For instance, we can hold the function constant in one variable and get two more variants of the function, \( \chi_m(n) \) and \( \chi_n(m) \).

Edit

Hint:Hold the variable \(m\) constant at zero, and consider the function \( \chi_0(n) \) . What possible value or values does this function produce on positive values of \(n\)? What possible value or values does this function produces on negative values of \(n\)?`There is no particular reason. For instance, we can hold the function constant in one variable and get two more variants of the function, \\( \chi_m(n) \\) and \\( \chi_n(m) \\). Edit **Hint:** Hold the variable \\(m\\) constant at zero, and consider the function \\( \chi_0(n) \\) . What possible value or values does this function produce on positive values of \\(n\\)? What possible value or values does this function produces on negative values of \\(n\\)?`

Cool! I'm gonna rename this post "Just for Fun 3". I don't mind other people sharing my numbering system on these "Just for Fun" posts because they're... just for fun!

`Cool! I'm gonna rename this post "Just for Fun 3". I don't mind other people sharing my numbering system on these "Just for Fun" posts because they're... just for fun!`