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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?

• Options
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?
• Options
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\$$? 
• Options
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!
The text passage contains multiple mathematical questions or puzzles, but no answers are provided for these questions. The puzzles are related to analyzing the properties and names of certain functions defined on integers.
multiple_questions
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
#### Howdy, Stranger!

It looks like you're new here. If you want to get involved, click one of these buttons!

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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?

• Options
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?
• Options
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\$$? 
• Options
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!
user
The text passage contains multiple mathematical questions or puzzles, but no answers are provided for these questions. The puzzles are related to analyzing the properties and names of certain functions defined on integers.

TYPE: MULTIPLE_QUESTIONS