Believe

I wrote this one when I was beginning to get too comfortable with teaching that I wanted to forego my graduate studies in UP. Unconsciously, I was building a comfort zone. I would justify to myself that taking a masters degree in Cebu while teaching 8 math subjects to an average of 480 students a week, is the selfless and most responsible thing to do. No matter how I justify it though, it all simplifies to one thing: I was scared shitless. I was scared of failure. I have wanted to study in UP since I was in high school and was afraid that I might realize that I was never good enough for my dreams. However, when it reached the point that I was slowly complying with the requirements of a graduate school in Cebu, alarm bells went off. And selflessness be damned. I ripped off the page where this poem was written, kept it in my pocket and decided to fly to Manila right after summer classes was over. I guess it was just appropriate that the last class I taught that summer was Mathematical Logic.

That was almost five years ago. I finished my masters degree already, passed SOA exams but can't seem to find the work I want. I am scared again; plus maybe, a little bored, depressed and hopeless. Pretty strong feelings, right? Yep, and negative and useless because when totalled, I am still jobless. I guess it's about time that I start believing again.

~0~0~

Move in tune

with the song

of the earth.

Dance with the

tulips and let

the sun admire

your grace

and the moon envy

your youth.

Believe Mae and

the wind, earth,

water and fire

will conspire with you,

with your dreams...

Of Techniques...

These three problems were presented during the Area Stage Competition of the 10th Philippine Mathematical Olympiad. On the average, each of them requires only a few minutes to solve, so if you find yourself working on one way too long, change techniques.

If 2A99561 is equal to the product when 3(523+A) is multiplied by itself, find the digit A.

The perimeter of a square inscribed in a circle is p. What is the area of the square that circumscibes the circle?

The sum of the first ten terms of an arithmetic sequence is 160. The sum of the next ten terms of the sequence is 340. What is the first term of the sequence?

Apples and Alex

OMG! When I read in yahoo news that Stephanie March will guest star in Law and Order: Special Victims Unit for six episodes, I squealed, shrieked and cussed! It's about damn time that they bring back the kick-ass Alexandra Cabot in the fold. Let's admit it, ADAs Casey Novak and Kim Greylek are not half as good as Cabot; hell, not even a quarter as good! They can and never will, fill in Cabot's shoes. Novak realized this early on and I bet Greylek will too, sooner or later. Or maybe she does already; I wouldn't know, I boycotted the show. I mean, I wasn't following every episode as religiously as I used to. I guess people will remember Novak as the ADA who replaced Alex Cabot and Greylek as the someone who replaced a replacement. What can I say? Cabot completes SVU; she sets the bar. Even the ratings are proof enough and numbers don't lie. It literally swan-dived after she was forced into Witness Protection Program. So after hearing that she will return for six episodes I am grinning rather widely it hurts! I guess saying that I am excited is honestly an understatement.

Ok, moving on. The problem below merits posting simply because we encounter it way too often. Well that, and the fact that I can't find the list of math problems I compiled a long time ago. Anyway, this problem illustrates the use of the technique known as working backwards. Read and try solving it and you will know what I mean. I can't help it, I just have to name the character, Alex.

Alex bought a bag of apples on Monday and she ate a third of them. On Tuesday, she ate half of the remaining apples. On Wednesday when she looked inside the bag, she found that she had only two apples left. How many apples did she have at the start?

Buddha in Glory

No, this is not a math problem. Yes, this is a poem and one of my favorites. Buddha in Glory is one of the greatest tribute to self-exaltation. What inspires me is the fact that the poem affirms the endless and continuous possibilities present to an enlightened man.

Buddha in Glory

by Rainer Maria Rilke

Center of all centers, core of cores,

almond self-enclosed, and growing sweet--

all this universe, to the furthest stars

all beyond them, is your flesh, your fruit.

Now you feel how nothing clings to you;

your vast shell reaches into endless space,

and there the rich, thick fluids, rise and flow.

Illuminated in your infinite peace,

a billion stars go spinning through the night,

blazing high above your head.

But in you is the presence that

will be, when all the stars are dead.

For J

The first time I read this problem it was my second-year in high school, a time when PhP20 can buy a lot and the Backstreet Boys mania thick in the air. The most recent time I encountered this problem was more than ten years hence, when I was reviewing for SOA Exam P and performing the statistical treatment for J’s thesis. In one of our discussions, J commented that it is impossible to determine whether a student who had identified the correct answer in a multiple-choice question really knew the answer and was not simply guessing. True, we can't say so with 100% certainty. But with certain assumptions, it is not completely insolvable either. The problem below illustrates this.

In answering a question to a multiple-choice test, a student either knows the answer or guesses. Let p be the probability that the student knows the answer and 1-p be the probability that the student guesses. Assume that the student who guesses the answer will be correct with probability 1/m, where m is the number of choices. What is the conditional probability that the student knew the answer to a question given that he had answered it correctly?

The answer is given by

We illustrate this with an example. Suppose you gave a multiple-choice exam with five possible choices. Student A whom you believed to have a 10% chance of knowing the correct answer for a particular question was able to identify the correct answer. The expression above tells us that the probability that Student A really knew the answer and was not simply doing "eeny, meeny, miy, moe" given that he has answered correctly is 5/14. In other words, the probability that Student A was just lucky is 9/14.

The Monty Hall Problem

The Monty Hall problem appeared on the movie 21, a film from Columbia pictures inspired by the story of the MIT Blackjack Team. During an advanced class, Professor Micky Rosa, played by Kevin Spacey, challenged Ben Campbell, played by Jim Sturgess, with the Monty Hall paradox which Campbell solves successfully. This meeting eventually led to Campbell joining the blackjack team headed by Rosa. Together with four other MIT students, the team went to sweep Vegas' casinos by storm using card counting. Below is a version of the Monty Hall problem as it appears on the book, Introduction to Mathematical Statistics (6th ed.) by Hogg, McKean and Craig.

Suppose that there are three curtains. Behind one of the curtains is a nice prize while behind the other two are worthless prizes. A contestant selects one curtain at random, and then Monty Hall opens one of the other two curtains to reveal a worthless prize. Hall then expresses the willingness to trade the curtain the contestant has chosen for the other curtain that has not been opened. Should the contestant switch curtains or stick with the one she has? What is the probability that he wins the prize if he switches?

To answer the first question, we only need to know the probability that the contestant wins the prize should he change curtains. Using the Law of Total Probability, this is given to be 2/3. This means that the odds in favor of the event, winning after switching curtains, is 2:1. Thus, it is advisable that the contestant switches.

Chapter 0

Q.E.D. is an abbreviation of the Latin phrase quod erat demonstrandum which literally translates to, that which was to be demonstrated. This is one of the oldest and most commonly used symbols displayed at the end of a mathematical proof or argument to indicate its completion. Other symbols include a solid square, a sharp, which I used throughout high school, and even the innovative letters, H.N.G., which means "human na gyud".

This blog will mostly contain some of the math problems that have sparked my interest at one time or another and will resemble the math calendar found in the Mathematics Teacher journal. I have no idea how to post the solutions just yet so if you want a copy, you may email me. I might also post some problems which I haven't solved, but they will be identified accordingly. Hopefully, this blog would push me to realize a compilation of math problems that I have been planning to work on.

I invite you to read on and recall your algebra, trigonometry, geometry, probability, calculus and other tricks, but unlike high school, have some fun (if possible). Prove, re-solve the problems in your own way (with or without listing the GIVEN, as probably required before), enjoy or simply be confused. Well, what can I say? One problem a day keeps the boredom away!