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Posted by tpc at May 18th, 2012

Have you ever watched rugby matches on TV and noticed the advertisements on the field that looked 3-dimensional? I remember having the impression that the adverts were superimposed by the TV people. But when I told the wife, she plainly pointed out that when the players ran across the field, their bodies covered the ads and not the other way round.

An example of the 3D ad (image from here.)

That was years ago but I was reminded of this when I read “How to Take a Penalty: The Hidden Mathematics of Sport” by Rob Eastaway and John Haigh. To achieve the 3D effect, the ads people made use of projective geometry and drew the ad in such a way that it looked like 3 dimensional. The best image I can find is this plan view from google maps. Notice how the top of the rectangular shape is longer than the bottom.

BTW, the book I mentioned is quite a good read.

Posted by tpc at May 7th, 2012

I’ll be visiting China tomorrow and my mum kindly handed me a stack of RMB (China’s currency) that was left-over previously. I took a quick count and it totals exactly 1729 Yuan. What are the odds?

Posted by tpc at March 14th, 2012

The New York Times published an article - which was re-published in the local papers - about a former mathematics professor Kim Myungho who self-published a book decrying the Korean judiciary. (Note the word “published” appeared thrice in the previous sentence.) In fact a movie already seen by several millions was made about Kim who contested a wrongful dismissal against his University.

That piqued me and googling “Kim Myungho professor” which incidentally was suggested by google, lead to this page which is a reprint of a letter published (that word again!) in the Math Intelligencer in 1997. Apparently everything began when Kim spotted an erroneous exam problem. Certainly some food for thought.

Posted by tpc at March 14th, 2012

A little creative tinkering yielded this mnemonic for Pi (apologies, latex died) to 15 places.

“How I need a drink, alcoholic of course, after the heavy lectures involving Euclid’s algorithm”

I’ve substituted “quantum mechanics” with “Euclid’s algorithm”, not exactly ideal since it is usually called Euclidean algorithm, plus do you count that apostrophe? But at least it is about maths and my students are doing number theory. However, it still needs to be tinkered with due to the mentioning of that certain beverage which can be construed as offensive to certain Asian sentiments (or values.) This is after all, not Australia!

So the revised version is:
“How I need a snack, chocolate of course, after the heavy lectures involving Euclid’s algorithm”

Posted by tpc at March 14th, 2012

The elementary theory of numbers should be one of the very best subjects for early mathematical instruction. It demands very little previous knowledge; its subject matter is tangible and familiar; the processes of reasoning which it employs are simple, general and few; and it is unique among the mathematical sciences in its appeal to natural human curiosity. A month’s intelligent instruction in the theory of numbers ought to be twice as instructive, twice as useful, and at least ten times as entertaining as the same amount of “calculus for engineerings.”

Taken from p.818 of his 1929 paper in the Bulletin of the AMS, the text of the sixth Josiah Willard Gibbs Lecture.

Posted by tpc at February 27th, 2012

There are two kinds of generalizations. One is cheap and the other is valuable. It is easy to generalize by diluting a little idea with a big terminology. It is much more difficult to prepare a refined and condensed extract from several good ingredients. — G. Polya

From Induction and Analogy in Mathematics, 1954.

But I actually saw it from the cutesy “marginal” comments from Concrete Mathematics.

Note that google books has the original Polya quote as

You should not forget, however, that there are two kinds of generalizations. One is cheap and the other is valuable. It is easy to generalize by diluting; it is important to generalize by condensing. To dilute a little wine with a lot of water is easy and cheap. To prepare a refined and condensed extract from several good ingredients is much more difficult but valuable.

Posted by tpc at February 26th, 2012

Came across an interesting problem Brent Yorgey from his blog The Math Less Traveled. In this post he tried to explain what is meant by a combinatorial proof to prepare his readers for a proof of this identity
… to be continued …

Posted by tpc at January 31st, 2012

Had a good laugh today reading about Justinsomnia syndrome. (via this rant about qr codes via JD Cook’s tweet.)

For me QR-codes are great because they are mysterious (at first) and students find them interesting. And I’m saying this after having actually tried it in class. I printed out the codes which had links to some resources and passed them to my students and asked them what those were? Some of them knew and told the others and I reckoned at least 30% of them used a smartphone to access the url and messages that I have created. (This was an undergraduate class and so the students aren’t barred from using their smartphones.)

On the other hand, a smartphone or in my case the i-pad is not so good for writing. Because the device is so convenient, I hardly power up my laptop at home any more. I do almost all my reading on the i-pad and that’s great but it’s just reading. I find it a chore to type on the ipad so there is practically no writing or blogging. And it took something as funny as the justinsomnia story to jolt me out of my blogging inertia.

Posted by tpc at December 14th, 2011

A general article on the Riemann hypothesis written by two insanely brilliant number theorists from Oz. It is part of a series explaining the millennium problems.

Posted by tpc at December 2nd, 2011

We have seen textbooks that only gives solutions to odd-numbered exercises. But have you seen a number theory text with the following exercises?

1) Prove that has no non-trivial solutions in integers.
2) Prove that has no non-trivial solutions in integers, for all

That’s actually apocryphal. Another story which appeared in George Dantzig’s obituary, tells of how he arrived late for a class one day to find two problems on the board. Thinking these were homework, he went back and solved them.

“The problems seemed to be a little harder to do than usual,” he said.

Turned out these two were open problems in statistical theory. (Note: I somehow remembered the protagonist as Paul Cohen.)

So perhaps it’s not too far-fetched to find the following exercise in Lang’s Complex Analysis. I checked my copy and it is a real-ly there!

Posted by tpc at November 20th, 2011

is a book I chanced upon in the library. The book consists, as the title states, of the constant to two million digits in 296 pages. My first reaction was, what is it doing in a university library? I’m all for esoteric number theory, and computing the digits of pi is as fruitful an endeavour as perhaps practising corporate law/finance. But in this internet age, why would you need a printed list of the digits? I can only imagine it’s because purchasing is done by people who do not actually read the books. Still one can find recreational uses for the book. See the reviews of Pi to five million digits - an expanded version - on amazon. For good books on pi, I personally recommend A history of Pi by Petr Beckmann and the resource book Pi: A Source book by Berggren, Borwein and Borwein. Not forgetting Borwein and Borwein’s Pi and the AGM. And why do all the last names begin with B?

Posted by tpc at November 20th, 2011

I’m one of those who first reaction is to “google it” when I want to check some fact. I even had a first generation SSD netbook which boots up in less than one minute to allow me to check things that occurred to me. Now with armed with an ipad2, it’s even easier.

It’s interesting that in the past week I read two pieces related to the accuracy of wikipedia. The first via J D Cook was how Dan Lemire’s wiki entry written in Dec 2010 was flagged for plagiarizing a magazine published in Mar 2011. The second piece is the following dig by xkcd.

Posted by tpc at November 12th, 2011

Singapore has very strict laws on drug trafficking. Anyone found with more than 15 grams of heroin faces mandatory capital punishment. I remember learning this fact in primary school. Even on Singapore Airlines flights into Singapore, this fact would be announced over the PA system to warn tourists or anyone entering the country.

According to Amnesty International more than 400 people were hanged in Singapore since 1991, mostly for drug trafficking offices. So it was strange to read in today’s newspaper about a ‘cobbler’ heroin trafficker who was jailed for 21 years. His role was to transfer the heroin from the shoes of three couriers to those of another three couriers. According to the report, 1.5kg of heroin was seized. That’s enough to warrant 100 capital sentences. Besides this cobbler, another courier has pleaded guilty, and both of them were charged with trafficking “not less than 14.99g” of heroin. I’m guessing there is some sort of deal being cut, where if you plead guilty you escape the noose. The remaining 5 are claiming trial.

Now if you approach this logically:”if you traffic more than 15g then you face the death sentence.” The contrapositive of the statement is “if you do not face the death sentence then you did not traffic more than 15g.” So “less than 15g” is equivalent to “not less than 14.99g”. Of course, there is actually no contradiction here because you could in fact be trafficking 14.999g.

Yes there is a pun in the title.

Posted by tpc at October 7th, 2011

is a short article rewritten by Thomas Korner about how students should view lectures in mathematics. It should be compulsory reading for all undergraduate maths majors. Note that his webpage is aptly named Korner’s korner.

Posted by tpc at September 26th, 2011

It can get you a job? In Mr Magorium’s Wonder Emporium, the first question the prospective accountant was asked was Name the Fibonacci series from its eleventh to its sixteenth. The accountant answered correctly and after another couple of questions he was hired.

Well that’s a movie. A real life application came from Turan, which I will quote from the book “My Brain is Open” by Bruce Schechter.

In 1935 Turan wrote a paper on a problem in number theory with Erdos, which appeared in the Russian Bulletin of the Institure of Mathematics and Mechanics of Tomsk. Ten years later, in postwar Budapest, Turan was stopped by a Soviet patrol … The soldier demanded that Turan produce his papers*. Turan had lost his ID a few days earlier evading another roundup, but reaching into his briefcase he found a copy of the paper he had written with Erdos. Turan handed the paper to the soldier, who, duly impressed that Turan has been published in a Soviet journal, let him go. Turan later dryly reported this story to Erdos as a “surprising application of number theory.”

* my added comment: as in ID

Posted by tpc at August 8th, 2011

A certain Count had 5 daughters and 11 peasants. Upon his death he willed that the square shaped county be divided into equal square plots such that each daughter gets one plot and the remaining plots are divided evenly among the peasants. Is there a solution?

Yes: Divide the county into 49 plots. 5 plots go to the daughters and the remaining 44 can be divided evenly.

Is there a solution if he had 2 daughters and 11 peasants? The answer turns out to be no and it lies in the fact that
has no solution.

The story is taken from Pommersheim et al.

Posted by tpc at August 3rd, 2011

One downside about writing calculus lecture notes has always been drawing graphs. They just take up too much time. What I used to do was to use whatever software at my disposal (usually powerpoint) and created jpeg to embed into my pdf. But it looked untidy and it is difficult to make slight changes to the jpegs.

Two colleagues have been singing the praise of pstricks, but I did not like having to latex -> dvi -> ps -> pdf. Luckily I discovered the PGF/tikz package which is another drawing package in latex. I found the PGF/Tikz system much more amendable, you could do latex to dvi or directly use pdflatex.

Of course it is still a chore to write latex commands. I get around it using geogebra which is a freeware for essentially geometry but it generates PGF/Tikz as well as pstricks code. Once the code is written, it is easy to make adjustments like adding labels, changing axes etc.

The downside of the PGF/Tikz is that certain graphs do not come out right. For that you would have to use gnuplot. It took me a few frustrating days to get everything to work properly in winedt. There are two key tweaks for using gnuplot with winedt on Windows 7. Many thanks to their respective contributors (and google for finding the postings.)

1) After setting the gnuplot path in windows environment, you also need to set it in winedt
http://comments.gmane.org/gmane.editors.winedt/6138

2) You need to enable write-18 for pdflatex to invoke gnuplot
http://www.politicaldata.org/?p=14

Posted by tpc at July 14th, 2011

my department (a.k.a academic group) loaned me a TI-84 Plus Silver Edition because students here are all supposed to be equipped with one. I just love how I’m learning new mathematics everyday with the TI-84. For example,

1) Andrew Wiles is wrong because according to my TI-84

2) Pi is a rational because according to my TI-84

I look forward to learning more new mathematics with my TI-84.

*just in case you do not understand sarcasm, I have provided the following references.
1) Please see the page on Simpson’s Maths Season 7 under episode Treehouse of Horror VI .

2) Martin Griffiths: The Mathematical Gazette, Vol. 86, No. 506 (Jul., 2002), pp. 263-264

To be fair, MATLAB tells me the same thing.

Posted by tpc at July 10th, 2011

I’ve never liked IQ tests. I had my share of SATs and Maths competitions when I was young but do not recall ever having my IQ checked. Having interacted with very smart people from over the world, I have my feet firmly planted when it comes to my own perception of my intellect. I’m average. Ok, maybe slightly better than average since I get to teach university students.

There are lots of pattern recognition questions and that’s my beef with them. The writers of IQ tests think of a certain pattern, and pronounce you “clever” if you are able to see that same pattern that they see. But who died to make them the arbitrator of patterns?

I think these tests are flawed and worse of all one can drill for it, just like drilling for the SATs and GRE. (It not uncommon for students who cannot speak a proper english sentence to do well in verbal sections of these tests.) To demonstrate this, I did a total of 3 tests, in a day. Each consisting of 50 questions. And my score went from a modest 20 to 29 to 31. According to the grading, I was of average IQ in the morning, good IQ in the afternoon, and of very good IQ in the evening. The difference? After one test, you figured out what the pattern recognition questions are asking for. A typical example is this, if there are 3 columns with diagrams. The third column usually is the union or the difference of the first two. Another typical example is when they are you to pick the picture that is the odd one out among five. There will be two pairs of identical ones and you try to single out the singleton.

The tests, by the way, were written by a pair of UK Mensa puzzle editors. And it is interesting that your IQ is higher if you were acquainted with American presidents according to these two Brits. There were 2 out of 150 questions on American presidents but none on any type of personalities from anywhere in the world.

Posted by tpc at July 10th, 2011

One of the little joys in the academic life is when you finally receive the hardcopy reprints of your published paper. I guess it is an antiquated practice dating back before the electronic era. In days of old, when manuscripts were handwritten, it would be wonderful to have copies of your own work typesetted professionally with which to send to other researchers in the field. Nowadays, chances are you would have sent a preprint by email to whomever you thought would be interested in the paper, even before you send it to a journal. With the typical lag of 1 to 2 years for the paper to be refereed, revised, appear electronically and finally appear in print, you would have in the meantime shared your work at conferences, corresponded with others and possibly moved on to another project.

So some of the journals like the journal of number theory have already done away with sending you free paper reprints or off-prints. They gave me a special authors electronic reprint with a cover page. Still, it is quite nice to have off-prints. Not the typical 50 copies but perhaps at least one for keepsake. I received my set of 50 offprints from Springer this month. Imagine my dismay when I opened the package to find that it was damaged by water somewhere during postage. The local postal company had a note to say it received the packaged already damaged and there is nothing I can do about it. Although the pages have dried up, they are warped and sticking to each other. It is much better to print off a new copy from the laser printer.