Writing Prompt Response

I’m going to attempt to write this in less than 500 words, here is the prompt:

[WP] Two kingdoms are at war with eachother. A third, larger kingdom becomes a threat to both of them. In order to defeat them, they must put aside their differences and work together. They don’t.

He watched them making their moves in battle, like grandmasters at the chess board. He watched on, clutching in his hands pieces of his own, mixing them in his palms anxiously.

Would they notice him, standing there, watching them?

The man on the left is absorbed in thought and eventually makes his move, a wing of cavalry enters the fray and cuts down out-of-position infantry. The man on the right glances in the stranger’s direction, but pays him no heed. He reaches out his hand and moves a piece. The stranger watches.

The man on the left takes an opposing piece and tosses it onto the floor. The man on the right reinforces his position. The stranger watches.

It had been years since battle had been joined, and neither side had ground out a winning advantage. A bloody stalemate approaches where neither side possesses the resources to continue. Still though, they play. The man on the left removes the last major piece from the board. The man on the right digs deep into his reserves and launches a hopeful counterattack. The stranger watches.

Hours passed, and finally, the two men sag in their chairs and reach their hands towards each other in a resigned handshake. Their pieces all scattered around the room, only their kings remaining on the board. The stranger strides forward, his time had come.

“Whom shall I play first?” The two tired masters look at each other in shock, but the man on the left vacates his chair a moment before the other. “Excellent.” The stranger sits at the table and sets up his pieces, a full army. The man looks around the room forlornly at his broken pieces, and places his king on its starting square, alone, staring at the horde staring back at him. The stranger reaches out his hand to move his pawn.

“Let us begin.”

Advertisements

On Improvement

A poem written in the Heathrow departure hall, my first attempt at some free verse.

On Improvement

I love to learn, to improve, but
How scared of criticism I am!
Why am I scared to compete and
Put my rating and ranking at risk?

The only way to become the best
is to participate and seek feedback
To put what you have done underneath a spotlight
and see if it has been found wanting

How scared I am to offer my opinion
On the topics of the day.
How scared that someone may disagree,
that is may be wrong or ignorant.

But sometimes I take solace that
wisdom is sometimes knowing what you do not,
But is that always so?
Does occasionally one need to choose a side?

A vow to be brave, I shall make,
to no longer be a comrade to fear.
To put my talents and work for others to survey,
That it may be improved and made stronger.

Different Forms of Poetry and Lyric Art

I’m attempting here to catalogue for myself some different forms of poetry so that I can start to write some!

My initial plan is to begin with some small poems and write numerous ones, I’m not going to focus so much on quality at the beginning, just quantity. I’m doing this so that I feel like I have gotten my feet wet and get over the hump of simply just beginning.

With no further ado, here we go!

Cinquain

This form goes:

2 syllables
4 syllables
6 syllables
8 syllables
2 syllables

It also says that the lines are usually iambic which means we have a sort of ‘da DUM’ rhythm with the stress on the off-beats.

Haiku

A Haiku is made up of 17 syllables with a 5/7/5 syllable count.

It is customary for a haiku to make references to nature, and is associated with a sense of place, where, when and what. They also recommend to not have a sentence per line, as this helps with the flow of the line.

Types of Meter

There are several different types of meter that are used, made up of the type of foot and how many are used in a line

Types of feet:

  • Iambic – A foot that has two syllables with the stress on the second syllable.
  • Trochaic – A foot with two syllables with the stress on the first syllable

Number of feet in a line:

  • Monometer – 1 foot per line
  • Dimeter – 2 feet per line
  • Trimeter – 3 feet per line
  • Tetrameter – 4 feet per line
  • Pentameter – 5 feet per line (iambic pentameter, sound familiar?)

Free Verse

Free verse poems are without meter or rhyme schemes or other rhythmic features. This is in contrast to blank verse which uses meter but no rhyme scheme, these are almost always written in iambic pentameter.

Sources:

https://www.poets.org/poetsorg/text/haiku-poetic-form

https://en.wikipedia.org/wiki/Iambic_pentameter

http://thepoetsgarret.com/list.html

http://www.writing.upenn.edu/~afilreis/88/meter.html

https://www.poetryfoundation.org/learn/glossary-terms/free-verse

Errors that students make in introductory statistics (Part 1)

It’s that time of the year again – final exams have rolled around and, as a tutor, I’m sitting around marking them. Exam marking can be an extremely depressing or entertaining process, depending on the state of mind and quality of the exams being marked.

I thought it would be interesting to document here my thoughts on the general things that people get wrong, and this can be instructive for students and teachers alike.

1.The Central Limit Theorem means that all samples are Normally distributed.

I had to start with the Central Limit Theorem, I’m not sure that there is anything less well understood as the CLT in undergraduate statistics. Let’s be clear, the Central Limit Theorem says that if our sample size is large, then the sample mean has a Normal distribution. The sample itself does not.

2.Using sample parameters in hypotheses

I see a lot students including things like

H_{0}:\bar{X}=50

and students do not seem to understand the significance of the hypothesis test. Doing this implies we’re checking to see if our sample mean is 50, this is easily checked by inspection, and not very interesting. Students who do this are failing to see the bigger picture, which is we’re using our sample to check if the unknown population mean is equal to 50, this is much cooler!

3.Not checking their understanding

Statistics is a branch of mathematics, which at its base is logic. If it seems like magic, or if it doesn’t make sense to you, I’d check your understanding. Ask questions and be curious, use your teacher’s office hours. I promise it will make sense eventually.

 

Failures of Instant Runoff Voting

I have always thought that Australia has had a perfect electoral system, at least for the House of Representatives. Compulsory ranked voting with instant run-off? What’s not to like?! The electorate is forced to show up once every four years and have their say, and no-one’s vote is wasted, or so I thought. I was recently made aware of two theorems in social decision theory, Arrow’s Theorem, and the Gibbard-Satterthwaite Theorem, that have made me more uneasy about our voting system, and voting systems in general.

Arrow’s Theorem

Arrow’s Theorem, or Arrow’s Impossibility Theorem, is stated as follows (Taken from here):

Let A be a set of outcomes, along with a process which aggregates individual’s preferences into a single preference order on A.

Whenever A has more than two outcomes, then at least one of the following must not hold:

  • (Weak Pareto Efficiency) If alternative a is ranked higher than alternative b for all individuals, then a is aggregately ranked strictly higher than b.
  • (Non-dictatorial) There is no individual such that that individual’s strict preferences always prevails.
  • (Independence of irrelevant alternatives) The social preference between x and y should only depend on the pairwise individual ordering of x and y. For example, adding a third candidate to the election should not affect the winner unless the third candidate wins.

Clearly having a non-dictatorial system is extremely important, though one could critique the power our media institutions have in choosing the winner of an election, and Weak Pareto Efficiency is also important. However, this usually means that an electoral system will fulfil these two properties, which means that the third property is usually unfulfilled. This means that in Instant Runoff Voting, we have situations where it may be advantageous to us to not enter our true preferences on the ballot. What type of situations could this occur in?

An Example

Well, let’s consider a scenario where we have the following first preference polling numbers:

Greens: 33% (33% Labor)

Labor: 30% (14% Liberal, 16% Greens)

Liberal: 37%

Suppose your favoured party is The Greens, followed by Labor, and then Liberal. Suppose also that you know that the preference distributions is as above. In the run-off, Labor will be eliminated, and the preferences allocated as follows:

Greens: 49%

Liberal: 51% (Winner)

and we have ended up with our least favoured outcome.

However, if instead 3% of Greens voters put Labor as their first preference instead, we get:

Greens: 30% (30% Labor)

Labor: 33% (14% Liberal, 19% Greens)

Liberal: 37%

and now The Greens will be eliminated, and the final results will be:

Labor: 63% (Winner)

Liberal: 37%

and we have ended up with our second choice by a landslide.

So in what situation will this come about? Well, as we can see above, it will happen when we have an extremely close election between three or more parties, and the preferences of the centre party are not sufficient to elect the minor party. It may be tactical then for the Greens voter to instead put Labor as their first preference.

You can see, however, that this requires detailed knowledge of both the first preference votes and the preference run-offs for each party, which is rarely the case. In the absence of such information, I would conjecture that it is a dominant strategy to put in one’s truthful preferences.

Alternative Voting Mechanisms

The previous section had me thinking, is there a way to bypass Arrow’s Theorem? It only holds for ranked choice ballots, right? Well, there are a plethora of other ways to vote! See Cardinal Voting among others, which does not use a ranked system, and therefore sidesteps Arrow’s Theorem. However, there exists a much more general theorem, the Gibbard-Satterthwaite Theorem, which says that for any voting mechanism, one of the following must hold:

  • There are only two alternatives,
  • (Non-dictatorial) There is no individual such that that individual can choose the winner,
  • (Existence of Tactical Voting) There is no action which best defends the individual’s desired outcome in every situation.

So it can be seen that there is no voting system which will satisfy all three of these simultaneously, and to me, that is a little bit troubling.

Voting photo taken from here.

 

 

Magic Squares (Part 1)

I was watching an episode of Penn and Teller Fool Us with my parents, if you haven’t watched this show, you really should. It’s extremely entertaining, though my girlfriend gets very upset that the show doesn’t tell you exactly how the trick is done.

In this episode, a magician by the name of Amazing Allison does a trick involving a magic square – you can see it here. When I saw it, I immediately wanted to know how it was done using mathematics.

A brief set-up: someone is asked for a number between 25 and 99, and then the performer creates a magic square, such that the rows and columns add up to the target number, and in addition a whole lot of other things add up to the target number, including the diagonals, each square of 4, etc. In my mind, there must be a formula for this, so I set out to find it.

Let us briefly consider a trivial solution: Suppose my target number is y, then a trivial solution for an n \times n magic square is for all entries to equal \frac{y}{n}. Of course, this is not a very interesting magic square, and quite easy to make! So let us improve this slightly by introducing a free variable – so now we have a target number y, and we fill in one box with our free variable x. The question is now, how will we find the remainder of our entries? Mathematically, this requires us to find n^{2} - 1 linearly independent conditions upon the entries of the square, then we chuck it into a matrix and find the reduced row echelon form, and whaboom we’re done. Easy, right? I tried this using a 4\times 4 magic square and it was surprisingly hard to find 15 linearly independent conditions! Here were all the conditions I added:

  1. All rows and columns must add to y
  2. Each diagonal must add to y
  3. The corners must add to y
  4. Each square of 4 blocks must add to y
  5. Each combination of non-corner exterior squares must add to y
  6. Diagonal alternating patterns through the square from top-down must add to y

and I was finally able to hit rank 15! The square yielded was as follows:

ScreenClip

Now, this is a good start, but looking at it, it still isn’t very interesting!

Going forward there are a couple of conditions that we should impose on this, firstly that each entry should be positive and integer. This will admit a trivial solution only in the case where y is divisible by 4. It may be better suited to a linear programming style problem, however, due to the integrality constraints. This will be further explored in part 2.

 

Partial Scoring for Multiple Choice Tests

I read some time ago an article by Terence Tao on how to assign partial credit scores for multiple choice tests – the article can be found here.

Terence’s solution focused on True/False questions, but could be easily generalized to multiple choice questions, but involved as part of the solution giving negative marks to students, including negative infinity marks. Clearly this is an infeasible solution for the classroom, and one wonders if there is a function f:[0,1] \to [0,1] such that f(0)=0 and f(1)=1 which satisfy similar conditions as those outlined in the article. The main condition that needs to be satisfied is that the students expected score should be maximised at their subjective probability of the particular answer being correct.

We can use some of Terence’s work here, to cheat slightly. Let us institute a rule that a person cannot bid 1 or 0 for a particular answer, and secondly that they can only bid a value q\in\{0.01,0.02,...,0.99\}. Now, we can use the form f(p)=Alog(p)+B with the constraints that f(0.99)=1 and f(0.01)=0. This yields the equation f(p) = 0.2176log(p) + 1.0022.

Let us check the required conditions:

f(0.01)=0: f(0.01)=0.2176\times log(0.01) + 1.0022 = 0

f(0.99)=1: f(0.99)=0.2176\times log(0.99) + 1.0022=1

E[mark] = p(0.2176log(q)+1.0022)+(1-p)(0.2176log(1-q)+1.0022)

\frac{\partial E[mark]}{\partial q}=0.2176\frac{p}{q}-0.2176\frac{1-p}{1-q}=0 so that the expected mark is maximised when p=q.

So we have found a function which practically maps to [0,1], with the minor caveat that a student can never pick either 0 nor 1 for their answer.

There is, however, a large flaw with this function, which is a student can lock in a score of 87% simply by choosing 0.5 for both options. Clearly further work is required here!