Nomic: Wiki Edition --- Finished (More or Less)

[StrangerCoug]

Election results are (in order of receipt):

Ircher's votes

1. StrangerCoug
   
2. Jake The Wolfie
   
3. Deimos27

StrangerCoug's votes

1. Ircher
   
2. Jake The Wolfie
   
3. Deimos27

Jake The Wolfie's votes

1. StrangerCoug
   
2. Gamma Emerald
   
3. Not_Mafia

That gives me 10 votes, Jake The Wolfie 6 votes, Ircher 5 votes, Gamma Emerald 3 votes, Deimos27 2 votes, and Not_Mafia 1 vote. I remain Head Director, Ircher remains on the board, and Jake The Wolfie is elected to the board.

Thank you for participating

[Jake The Wolfie]

(:

[Ircher]

Welcome to the board Jake!

[Jake The Wolfie]

I am a god of Chaos

[lendunistus]

boop

[Ircher]

The election is over, so StrangerCoug can post the results when he gets a chance.

Question of the Day #14

Category: Chinese
Question: The vast majority of Chinese characters have two parts: a semantic part and a phonetic part. What is the term (in English) given to the semantic part of these characters?

The correct answer is it is called the

radical

.

Question of the Day #15

Category: Video games
Question: Who is the main playable character in the series of games

The Legend of Zelda

?
---
(Also, don't forget to pm me answers to my quiz in 545 by Monday morning.)

[StrangerCoug]

Didn't notice the Chinese question. Should've gotten that one ><

Question: Who is the main playable character in the series of games

The Legend of Zelda

?

Link

[Ircher]

That is correct!

Give StrangerCoug 30 Knowledge Points.

[Jake The Wolfie]

The Hero of Time

[Deimos27]

Now that we got a sixth player I might end up going inactive due to low wim
In general I'm unmotivated recently for non-dancesport related stuff cause I was training for a competition I had last weekend and I have another next weekend and it's taking up all my mental real estate
Will keep an eye out occasionally for when win conditions might get implemented

[Deimos27]

competition I had last weekend

and by this I mean yesterday

[Ircher]

Good luck at your competitions!

[Ircher]

Consume 1 Victory Quiz Token.

Victory Quiz #1

Spoiler:

Directions: Answer the following questions to the best of your ability. You will not be penalized for guessing. Any fully correct answer will receive full credit regardless of any underlying work.

This quiz is open notes and open Internet subject to the following restrictions: 1) you may not use sites like Chegg, Quora, and StackExchange where other people answer questions 2) you may not directly search the provided questions 3) websites like Wolfram Alpha are also prohibited; see the policy on calculators listed below 4) apps of any kind (especially ones like Photomath) are also prohibited. You may look up words like "domain", "derivative", etc., but typing the full question into Google is not appropriate. Examples of acceptable use of the Internet include sites like Wikipedia, Desmos, Overleaf, and pages from a university. You are not allowed to consult with other people on this quiz. Calculators (including online calculators) are acceptable provided that they cannot perform symbolic computations (i.e.: they do not have a CAS). The only allowed programming language on this exam is R. Programs like Mathematica and Maple are also prohibited due to violating the calculator policy.

This quiz consists of

5 questions

and a maximum score of

100 points

. Show your work for partial credit. Unless otherwise specified, all answers should be given in an analytic form; that is, decimal approximations are not correct answers. The dollar signs delineate LateX code; you may want to type it into a LaTeX renderer like Overleaf. If you have any questions regarding the rules of this quiz, please ask.

Please remember that you are PM'ing me your answers. You have until (expired on 2021-11-22 10:00:00) to submit your responses.

Question #1 (14 Points)

: What is the domain and range of $f(x) = x ln(x)$?

Question #2 (8 Points)

: What is the derivative of $g(f) = f^2 + e^{\pi} \ln(\sqrt{17})$?

Question #3 (23 Points)

: A group is a set $G$ together with a binary operation $\cdot$ typically called multiplication that satisfies the following properties: 1) G is closed under multiplication 2) multiplication is associative; that is, $a(bc) = (ab)c$ 3) there is an identity $e$ in $G$ such that $ae = ea = a$ for all $a$ in $G$ and 4) for each element $a$ in $G$, there is an element $a^{-1}$ in $G$ called the inverse of $a$ such that $aa^{-1} = a^{-1}a = e$. We define the order of a group as the number of elements in the group, and the order of a group element is the number of times the element must be multiplied by itself to reach the identity. If no such number exists, we say that the order of the group element is infinite. Which of the following statements about groups and their orders are true?

(Please note: you may NOT directly search any of these statements.)

I) The set of integers under addition modulo $n$ form a group.
II) For all elements $a$ and $b$ in a group, $ab = ba$.
III) The identity element in a group is unique.
IV) For elements $a$ and $b$ in a group, $(ab)^{-1} = a^{-1} b^{-1}$.
V) The set of real numbers under multiplication form a group.
VI) A group of order 31 can have an element of order 6.
VII) The order of the identity is 1.
VIII) Let $a$ be a group element. If $a^k=e$ (that is, multiplying $a$ by itself $k$ times) for some natural number $k$, then the order of $a$ is $k$.
IX) Every group has an element of order 1.

Question #4 (25 Points)

: What words and expressions should go in the blanks in the following proof to correctly prove the vector Thales theorem: if

s

and

v

lie on one line through

0

,

t

and

w

are on another line through

0

, and

w

-

v

is parallel to

t

-

s

, then

v

= a

s

and

w

= a

t

for some scalar a.

(Notes: Boldfaced font indicates a vector. Each blank indicates a single word or a single expression. An expression in this context is any sequence of symbols that includes no equals sign. Answers should be given inline with some kind of separator (e.g.: color, blanks). You may search the phrase "vector Thales theorem", but you may not copy parts of the skeleton proof below.)

Proof:

Since

w

-

v

is ________ to

t

-

s

, there exists a scalar a such that

w

-

v

= ________ = ______. Then, since

v

is on the same line as ____, _____ for some scalar b. Likewise, since

w

is on the _____ _____ as

t

, there exists a scalar c such that

w

= ______. Thus,

w

-

v

= ______ and

w

-

v

= c

t

- c

s

. It follows that (c - a)

t

+ (a - b)

s

= ____. Since ____ and

t

point in ______ ______ from

0

, they are _______ _______. Thus, c - a = 0 and a - b = 0. Therefore,

v

= a

s

and

w

= a

t

. QED.

Question #5 (30 Points)

: Suppose $Y$ is a normally distributed random variable with mean 40 and variance $\sigma^2$. To the nearest integer, what value of $\sigma$ makes the probability that $Y$ is between 20 and 60 equal to 50%?

---
Don't forget: you can receive up to 100 KP and 3 points if you do well. (Also, even if you have no clue, please submit something. There is a good chance you will get something for your efforts no matter how small.)

Well, time is up and I didn't receive any submissions, so I guess everyone got a zero.

Anyway, the answers to this quiz are in the spoilers below:

Spoiler: Answers

Victory Quiz #1 Answers and Grading Rubric

Correct answers without work receive full credit.

Question #1 (14 Points)

: What is the domain and range of $f(x) = x \ln(x)$?

Answer:

The domain is all positive real numbers. To find the range, first find the relative maximum and minimum of $f$ by using the derivative:
$$f'(x) = \frac{x}{x} + \ln(x)=1+\ln(x).$$
To find the critical points, set the first derivative equal to 0:
$$1+\ln(x)=0\implies \ln(x) = -1\implies x=e^{-1}.$$
To determine if this is a maximum or minimum, use the second derivative test (the first derivative test also works):
$$f''(x) = \frac{1}{x}, f''(e^{-1})>0.$$
Hence, $x=e^{-1}$ is a relative minimum. Finally, determine the end behavior of the function; as $x$ approaches $0$, $f(x)$ approaches 0, and as $x$ approaches $\infty$, $f(x)$ approaches infinity. Hence, the range is $[-e^{-1}, \infty)$.

Scoring:

1 Point for knowing the difference between the domain and range.
1 Point for the correct domain.
3 Points for finding the correct derivative.
3 Points for finding the critical points of the function.
2 Points for determining the relative maximums and minimums of the function.
2 Points for determining the absolute maximum of the function.
2 Points for determining the absolute minimum of the function.

Question #2 (8 Points)

: What is the derivative of $g(f) = f^2 + e^{\pi} \ln(\sqrt{17})$?

Answer:

Notice that $e^{\pi}\ln(\sqrt{17})$ is just a constant, so its derivative is zero. Thus, $g'(f) = 2f$.

Scoring:

Approach 1: Use the limit definition of the derivative.
1 Point for writing the limit definition of the derivative.
1 Point for properly substituting the function into the limit definition.
4 Points for performing the algebra correctly.
2 Points for properly evaluating the limit.

Approach 2: Use derivative rules.
4 Points for recognizing that $e^{\pi}\ln(\sqrt{17})$ is a constant.
1 Point for properly differentiating $e^{\pi}\ln(\sqrt{17})$.
3 Points for properly differentiating $f^2$.
-1 Point if the product, chain, or quotient rules are attempted incorrectly.

Question #3 (23 Points)

: A group is a set $G$ together with a binary operation $\cdot$ typically called multiplication that satisfies the following properties: 1) G is closed under multiplication 2) multiplication is associative; that is, $a(bc) = (ab)c$ 3) there is an identity $e$ in $G$ such that $ae = ea = a$ for all $a$ in $G$ and 4) for each element $a$ in $G$, there is an element $a^{-1}$ in $G$ called the inverse of $a$ such that $aa^{-1} = a^{-1}a = e$. We define the order of a group as the number of elements in the group, and the order of a group element is the number of times the element must be multiplied by itself to reach the identity. If no such number exists, we say that the order of the group element is infinite. Which of the following statements about groups and their orders are true?

(Please note: you may NOT directly search any of these statements.)

I) The set of integers under addition modulo $n$ form a group.
II) For all elements $a$ and $b$ in a group, $ab = ba$.
III) The identity element in a group is unique.
IV) For elements $a$ and $b$ in a group, $(ab)^{-1} = a^{-1} b^{-1}$.
V) The set of real numbers under multiplication form a group.
VI) A group of order 31 can have an element of order 6.
VII) The order of the identity is 1.
VIII) Let $a$ be a group element. If $a^k=e$ (that is, multiplying $a$ by itself $k$ times) for some natural number $k$, then the order of $a$ is $k$.
IX) Every group has an element of order 1.

Answer:

I, III, VII, and IX are true. II and IV are only true in Abelian (or commutative) groups. The correct version of IV is $(ab)^{-1} = b^{-1} a^{-1}$. V is false because $0$ does not have a multiplicative inverse. VI is false by Lagrange's Theorem; the order of an element in a group must divide the order of the group. A counterexample to VIII is $e^2=e$. The order of the identity is 1, but the value of $k$ here is 2. It is however true that if $a^k=e$ for some natural number $k$, then $k$ divides the order of $a$.

Scoring:

2 Points for each true statement correctly marked as true.
1 Points for each false statement not marked as true.
1 Point for correctly marking all of I, III, VII, and IX true.
1 Point for each answer with a reasonable rationale (even if wrong).

Question #4 (25 Points)

: What words and expressions should go in the blanks in the following proof to correctly prove the vector Thales theorem: if

s

and

v

lie on one line through

0

,

t

and

w

are on another line through

0

, and

w

-

v

is parallel to

t

-

s

, then

v

= a

s

and

w

= a

t

for some scalar a.

(Notes: Boldfaced font indicates a vector. Each blank indicates a single word or a single expression. An expression in this context is any sequence of symbols that includes no equals sign. Answers should be given inline with some kind of separator (e.g.: color, blanks). You may search the phrase "vector Thales theorem", but you may not copy parts of the skeleton proof below.)

Proof:

Since

w

-

v

is ________ to

t

-

s

, there exists a scalar a such that

w

-

v

= ________ = ______. Then, since

v

is on the same line as ____, __ = ___ for some scalar b. Likewise, since

w

is on the _____ _____ as

t

, there exists a scalar c such that

w

= ______. Thus,

w

-

v

= ______ and

w

-

v

= c

t

- c

s

. It follows that (c - a)

t

+ (a - b)

s

= ____. Since ____ and

t

point in ______ ______ from

0

, they are _______ _______. Thus, c - a = 0 and a - b = 0. Therefore,

v

= a

s

and

w

= a

t

. QED.

Answer:

Since

w

-

v

is __parallel__ to

t

-

s

, there exists a scalar a such that

w

-

v

= _a(

t

-

s

)__ = __a

t

- a

s

__. Then, since

v

is on the same line as _

s

_, __

v

__ = __b

s

__ for some scalar b. Likewise, since

w

is on the _same_ _line_ as

t

, there exists a scalar c such that

w

= __c

t

__. Thus,

w

-

v

= __a

t

-a

s

__ and

w

-

v

= c

t

- c

s

. It follows that (c - a)

t

+ (a - b)

s

= _

0

_. Since __

s

_ and

t

point in __different__ __directions__ from

0

, they are __linearly__ __independent__. Thus, c - a = 0 and a - b = 0. Therefore,

v

= a

s

and

w

= a

t

. QED.

Scoring:

3 Points for a diagram/image of the setup.
1 Point for each correct blank.
2 Points for four correct blanks.
2 Points for eight correct blanks.
1 Point for twelve correct blanks.
1 Point for sixteen correct blanks.

Question #5 (30 Points)

: Suppose $Y$ is a normally distributed random variable with mean 40 and variance $\sigma^2$. To the nearest integer, what value of $\sigma$ makes the probability that $Y$ is between 20 and 60 equal to 50%?

Answer:

First, compute the normalized z-score associated with $20$:
$$z_{20}=\frac{20-40}{\sigma}.$$ Since the normal distribution is symmetric and the mean of $Y$ is 40, the probability that $Y$ is between 20 and 40 should be the same as the probability that $Y$ is between 40 and 60. Let $X$ denote a normally distributed random variable with mean 0 and variance 1. Using a calculator, we can compute the value $z$ such that the probability $X$ is less than $z$ is 25%. (The value 25% comes from the fact that the normal distribution is symmetric about its mean.) This comes out to be about -0.674. Hence,
$$\frac{20-40}{\sigma}\approx -0.674.$$
Solving this equation for $\sigma$ yields $\sigma = 30.$

Scoring:

5 Points for computing the normalized z-score.
6 Points for noting the normal distribution is symmetric around the mean.
14 Points for finding the z-score cutoff.
2 Points for setting the z-score equal to -0.674.
2 Points for solving for sigma.
1 Point for correctly rounding to the nearest integer.

Hopefully the next one I post (which won't be for a while) will be easier.

[StrangerCoug]

Mod: Please note sig.

[Jake The Wolfie]

Proposal 347: A player immediately wins if they are the only active player in the game.

[Ircher]

Buy 1 Swamp

Proposal 347: Tapped lands become untapped after 3 days.

We had a 347. Sorry, I haven't updated the wiki in a while.

[lendunistus]

boop

[Ircher]

Proposal 347 fails. Aronis is past the 3 day mark and is now considered inactive.

Motion M010: While this motion is active, rule 307 is suspended. This rule is active until November 29, 2021 11:59 PM UTC or for one second, whichever is longer.

[StrangerCoug]

VOTE: Yea M010

[Deimos27]

Welp looks like I'm not the only one with low wim
I do recommend implementing some win conditions
I do not care enough atm to attempt to think about how to do so myself sadly

[Deimos27]

Good luck at your competitions!

Thank you I am aiming to win at least waltz or quickstep

[Ircher]

(You should vote yea to M010 so we can enjoy a Thanksgiving reprieve.)

[StrangerCoug]

Prodge, not out of V/LA yet.

[Jake The Wolfie]

(You should vote yea to M010 so we can enjoy a Thanksgiving reprieve.)

(no.)

[StrangerCoug]

Mod: Out of V/LA.