Interesting article that I found while browsing Reddit math:

## New Layout

### Status

I’ve been experimenting with the new Showcase template and this is the result.

Some changes include additional widgets on the sidebar, the addition of “Recent posts” on the bottom of the homepage and linked “Featured Images” appearing on the “Featured” posts bar.

Thanks goes to Gaurav Tiwari and his blog, My Digital Notebook (link on the sidebar), where I got some of the ideas for the new layout.

# Fish and squares.

Currently I am working on two different problems, both of which I have added to the Current and Future projects page.

To summarize, these are…

# 1. A Fish Population Problem started by Gaurav Tiwari and his blog “My Digital Notebook”. A link to the post can be found here. A summary of the problem is as follows:

Let $F_{m}(t)$ and $F_{f}(t)$ denote the number of male and female fish, respectively, inside a tank at time t (where t is in a unit of days) and let $F(t)=F_{m}(t)+F_{f}(t)$ denote the total number of fish at time t. Initially we have $F_{m}(0)=4$ and $F_{f}(0)=3$.

Here are the conditions of the problem: A male fish can mate with any number of female fish, unless the female has already mated before. Female fish will always give birth to new offspring after 21 days of conception, with the first female always giving birth to 100 eggs, the second fish, 110 eggs, and the third, 90 eggs.

The new offspring fish will reach sexually maturity after 30 days and in each new batch, 60% of the offspring fish will die before reaching maturity. In the remaining 40%, the ratio of male to female fish is 3:2.

This second generation of fish have the following restrictions. Fish born of the same mother may note mate with each other and as with the first generation, a male fish can mate with any number of female fish, unless the female has already mated before. Every three females who give birth will always lay the eggs in the following order (from first fish to last): 100, 110, and 90. In these batches of eggs, only 40% will survive, with the new ratio of male to female being 1:1. In subsequent generations, the same restrictions found in the second generation of offspring will also apply.

Find the value of $E\left[F(365)\right]$, that is, the expected total population of fish found after one year has passed.”

# 2. A geometry problem that was started by a friend of a friend taking a Calculus III course at the University of  McMaster. Currently I am trying to find:

…how to calculate the volume of the solid displaced when a square plate of side length 2 is symmetrically passed through a sphere or radius 2 (the center of the square is threaded through a diameter line in the sphere).

as well as trying to attempt to solve the integral that represents said volume.

Today I have made considerable progress in both problems (for 1, see here, and for 2, see  here), but now I am at an impasse. This is why I would like to ask you, the readers, to give me some inspiration and ideas here but helping me answer the following questions:

1. For project 1, what suitable restrictions should be made so that the population can naturally be imbued with a carrying capacity?

2. For project 2, what would be a simple and intuitive way to integrate the $\arcsin$ portion of the integral.

3. For project 2, is there any way to re-write the integral to make the integration simpler?

# Future Plans and Projects.

While I am still quite busy preparing for Exam FM (completing 10 mock exams), I do have quite a few short-term and long-term plans for this blog.

For the sake of being concise, here is a short list of what I have planned; many of the projects will only be started after exam FM which is on the 10th of December. However, for some of these project, I might have some to time to start on them this week:

# Short-term Projects

• Finish my proof of the volume displaced from pushing a square through a sphere; an incomplete version can be found here.
• Complete the third entry in the high school reflection series
• Solve an interesting population problem about fish breeding here. I will also be re-typing the problem for better coherency (new version here)

# Long-term Projects (>3 months)

• Begin typesetting notes for Exam MLC. These notes will be extremely concise and should be considered a supplement for the recommended readings here. One of the texts is free while the other requires purchase (I acquired an eBook version). Since this is the first exam that will follow the new syllabus, I will be covering the content quite thoroughly with what is listed on the syllabus.
• Course notes for second and third year courses at uWaterloo (specifically the ones that I will be taking next term
• “Advice for the university freshman” series; look for it during the Winter term

# Committed Projects (>1 year)

• Ray Tracer project (currently on hiatus until other projects are taken care of)
• Martingales and Black-Scholes tutorial project (essentially a 30+ series of posts intended to get readers familiar with mathematically rigourous financial concepts)

In addition, I will be posting random tidbits of interesting maths finds here and there, but just don’t expect them to come that often during these coming couple of weeks.

## High school life, research topics, and advice for the young aspiring mathematician. (Part 2)

### Featured

This post will be a continuation of the content found in Part 1 which can be accessed here. In Part 3, I plan to end with a list of topics for those who are interested in writing an Extended essay in mathematics as well as some information for those who wish to study maths in university.

Specifically, this post will cover:

• The process of writing a maths essay versus a maths contest
• The most important characteristics of an aspiring mathematician

Recapping from part one, I was just taking my first couple steps into rigourous mathematics through an extended that was to be written on a mathematical topic independently for the IB program diploma. The only trouble was that I had no idea what to choose for my thesis and I could not depend on my adviser to guide me through picking one because he mainly worked in physics.

The only out-of-high-school mathematics topic that really stuck in my head during that time was Cantor’s set theory and cardinality concepts, which was fresh in my mind because I recently attended a workshop on them during my visit to CMS maths camp. Mainly his diagonal argument bijecting the natural numbers to the rational number was what really interested me.

... and Rubiks cubes. We had a lot of talented Rubiks cube players. But who would ever write an essay on Rubiks cubes anyways? {acobox.com}

Going on this small hunch that this topic could lead to a potential thesis question, I headed down to the library, read set theory books and browsed the internet hoping to come upon something interesting. However, most of the books and content that I did get a hand on (about 80%) was far beyond the scope of what was taught in the high school curriculum, so I did what any reasonable young mathematician would have done. I borrowed some books about how to read mathematics books.

This took about a week to really get the notation and symbols such as $\forall, \exists, \bigoplus,$ and \ vs / which you normally would see in only first to second year university mathematics. Once I got up to speed reading the set theory related books though, I decided to choose the broadest thesis that would accompany all the knowledge that I gathered so far. Something to the tune of “How could one construct a system for classifying the cardinality of any given set based on how the set is constructed?” what was immediately came up.

What followed was an entire year to construct an answer to my thesis, all under the constant pressure of course work and mandatory community service. But to me this didn’t matter, as long I spent more than a couple of minutes each day pondering about the solution, this kept me going and interested. New ideas would spring up constantly and unexpectedly because of this and sometimes even in my dreams! What kept me moving was living the experience and looking at what would eventually be there at the end.

Kind of like running a marathon, except without the burning sensation in your legs and lungs {acobox.com}.

4000 words and 21 pages of mathematics later, I completed my first draft of my paper in July 2009. I brought it to my adviser who was amazed at all the new concepts that he had never seen before and praised me for going the extra mile for keeping the maths simple and succinct.

Here I would like thank Mr. Vaithilingham (my adviser) for his guidance and support, offering advice about the format and alternate languages I could use to write my paper (this was where I had my first exposure to LaTeX). Although he did not go into the subject matter, he was kind enough to provide me with sample papers and offer me advice about what criteria my paper should fulfill.

In September 2009, I submitted my paper and in January 2010, I received the results. I received an A, the first one ever in the history of my high school. But while the result certainly made me happy, I felt that the effort and time that I put into writing my paper were the true rewards.

So what advice could gained from my long-winded story for the young mathematician? Well for that, we have to visit another one of my experiences in high school (last one, I promise).

Before, in the first part of this series, I mentioned that I wrote the Grade 10 Cayley Waterloo contest in grade 10. Well, before ending my high school career, I also took three other contests, the Fermat (for Grade 11) in grade 11 and the Euclid contest (for Grade 12) twice in grade 11 and 12. I can say that all four experiences were many distances apart from the process of writing a maths paper.

For example, I can guarantee you that there is no one way to effectively study for these contests other than perhaps reviewing your grade 11 formulas (no calculus was necessary). But with only pure memory skills, one can expect to get about 40% on the Euclid or 80% on the Fermat at best.

What is really necessary in these contests is a sharp intuition and the ability to apply shortcuts, shortcuts and more shortcuts. This is because contests are structured to test how quickly a contestant can blitz through the first half and apply critical thinking skills and deduction to the later half. This is especially important in the Euclid, where the general consensus (among my peers who have written Euclid) is that the first six questions should be written within the first 30 minutes while questions seven to ten should be done within the next two and a half hours; they’re that challenging.

"I knew I shouldn't have spent those thirty minutes trying to remember the area of a tetrahedron!" {CSULA}

This is where you can see the line drawn between the two activities. While writing a paper requires endurance and perseverance, a contest is more based on quick intuition, spontaneity and a sharp creative mind. But which of the two is more important? While having all these traits is necessary to become a successful mathematician, I personally believe that the former list outweighs the latter in terms of importance.

Why, you ask? Well, the hard fact is that mathematics is not easy. In fact, many of the problems that you will encounter in your graduate or even undergraduate career will not be solved within a day or even two. As an example, Andrew Wiles, the famous mathematician who proved Fermat’s Last Theorem, spent nearly ten years on that proof alone, which is a great testament to his commitment.

I found out about this first hand when I first entered into the “Advanced” mathematics classes in the University of Waterloo. Suddenly, problems on assignment because mini projects often taking several days to solve. It was very different to the finish-in-one-day assignments that I found in high school and part of first year university.

As further reinforcement, the author of “A Mathematician’s Survival Guide: Graduate School and Early Career Development” (a book that I would highly recommend) echoes these sentiments. In one chapter he describes a student, a young prodigy, who excelled in all his upper year maths courses, aced Putnam contests (the university version of the Euclid) regularly and was accepted to all the top graduate schools in the country. Yet, when he finished his graduate thesis, earning his Ph.D. the young student was

"so disgusted with mathematics that he quit and went to work in a
genomics laboratory"

So what happened? Well the author goes to explain that the student, during his undergraduate years, was always on the “Putnam Exam mindset”, always expecting to receive quick results without any hard work. He just didn’t have the tenacity or discipline required to do graduate studies.

Trying hard, persevering, these are the necessary things required to succeed in not just mathematics but in any subject that you study.

## High school life, research topics, and advice for the young aspiring mathematician. (Part 1)

### Featured

Brace yourselves as this is going to be a very long post – split into two to three parts – of myself reminiscing back to my high school years.

Specifically it will cover my experiences in:

• Grade 9,10 and part of 11
• My developing interest in mathematics
• The beginning of my Extended Essay on set theoretical topics

[Part 1]

To recount my experiences in high school, I can honestly say that for my first two years doing mathematics at the ninth and early tenth grade level that I really hated it, or at least hated how it was taught. Many of the problems were built around repetition and memorization and the “interesting” problems that were brought up in class were nothing more than algebra problems that were sugar-coated with words.

Problems like “if Alice is twice Bob’s age fifteen years from now and she is twenty right now, how old is Bob?” populated this category endlessly. I would often solve such problems and then write something to the tune of “Above is an algebra problem. To find Bob’s age, ask Bob for his health card.” just for kicks… and then proceed to lose marks because of it.

Because of the monotony of class, I eventually developed a system wherein I would be able to learn the material in the shortest amount of time while having free time to work on other subjects. Sort of a pseudo-form of procrastination if you will. I would sometimes even spend time reading ahead to the grade eleven and twelve books. Now, while I would always receive 95%+ on the tests, because I did not do any homework or in-class assignments, my marks would never peak beyond 85%. This frustrated me quite a bit.

"You may have solved the Riemann hypothesis, but you haven't completed your exercises on exponents yet. Therefore, you get a B- at best." {freedigitalphotos.net}

By the middle of the tenth grade my system still worked, but besides working on other class material during maths classes I was still bored out of my mind. It was only on a whim that I entered the Grade 10 Waterloo Cayley mathematics contest, or maybe it was out of boredom (I can’t recall). But taking that maths contest was possibly the best decision that I made up to that point in my life. This is because after writing it, not only did I win the contest and become the school district champion, the experience lit a flame within me whose goal was only to discover how far into mathematical ocean could I dive. All of this because the contest really challenged me and provoked me to improve myself to sheer effort.

"And also to allow for another chance to get back at those pretentious math club members and their non-mathy maths games." {freedigitalphotos.net}

In grade eleven, when I began driving straight into learning rigourous mathematics, I also began my two-year journey through the arduous IB (International Baccalaureate) program. One of parts of the program that I genuinely enjoyed was writing the Extended Essay, and in my case it was on Mathematics, specifically Cantor and his classification of the different cardinalities and properties of the cardinality of the continuum.

This was what essentially shaped my two years writing that essay... what else could be more fun? {wikipedia.org}

Now during the introduction of the essay and its components, which were taught through a series of lectures, my advisers made it very clear that no one chooses a maths topic unless they wanted a low grade or wanted to write about poker (which is what the last guy did… two years ago). Throughout the lectures, the content mainly emphasized skills that would be helpful for those writing history or English papers. None of it was particularly helpful for researching and formatting maths papers.

When it came time to select subjects and topics, sure enough, many of my classmates fell into either English, History or the odd Science related topic. When it came to my turn though, I shyly squeaked out my interest in developing a maths paper and instead of an expected “That’s a great choice.” or “What an interesting subject.” like all my other peers, I received a response to the tone of “Hah. Good luck with that. Nobody ever chooses math.” and yes those were exact words. No else in my class decided to do a maths paper after that.

"He wants to learn math? Hah! Next he'll be saying he wants to become a mathematician." {freedigitalphotos.net}

Undeterred though, I was hopeful that my personal faculty adviser could provide input on my thesis topic. Unfortunately, my maths adviser had never done any set theory during his undergraduate career and while that did surprise me a little I later found out, a week into researching my topic, that his majored subject was in physics and not mathematics.

I knew, then, that I had a lot of work ahead of me.