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Just For Fun 3

Keith E. PetersonKeith E. Peterson
edited April 2018 in Baez ACT 2019: Online Course

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.
    Dan SchmidtDan Schmidt
    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.
    Keith E. PetersonKeith E. Peterson
    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.
    John BaezJohn Baez

    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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