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# Just For Fun 3

edited April 2018

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?

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1.
edited April 2018

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?

Comment Source: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?
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2.
edited April 2018

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 $$n$$?

Comment Source: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\$$? 
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3.

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!

Comment Source: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!