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Just For Fun 3

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

  • 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?
  • 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\\)?
  • 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!
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