An identity

The following problem is apparently a bonus question for 13 year olds at a local girls school: Evaluate the sum
[tex] \displaystyle \frac{1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4} + \cdots + \frac{1}{99} – \frac{1}{100}}{\frac{1}{1+101}+ \frac{1}{2+102} + \cdots + \frac{1}{50+150}}.
[/tex]
It’s not that easy if you ask me. I had to work out the following identity first before I managed to solve it.
[tex]\displaystyle \sum_{k=1}^N \frac{1}{2k-1} – \frac{1}{2k} = \sum_{k=1}^N \frac{1}{k+N}. [/tex]
The identity can be proved via induction.

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3.14.15 and Einstein

14 March 2015, was supposed to be the Pi day of the century — for obvious reasons. 14 March is also Einstein’s birthday, and 2015 interestingly marked 100 years of the theory of relativity. I liked this article written by Jeff Edelstein that described how Einstein tutored a 12 year old girl in maths. As Edelstein wrote, this could be a hoax, but personally for me, some stories (or myths) are worth retelling. There are two lovely quotes in the article, both recollections of the girl being tutored.

He’d say we’re not going to bother with the homework problem. First he’d give me his own math problem, help me work it out, and then we’d go to the homework. He’d simply tell me the things we were doing, showed me the best ways to do it. And he would always tell me it’s not the answer that counts, but how you solve the problem.

But he was just so nice. Even when I told him I hated math. He said, ‘you shouldn’t hate math, math is the center of the universe, and anyone who knows math knows everything.’

Two more titbits about the great man. My paper with Hirschhorn appeared this year, and thanks to Mike — M.V. Subbarao — Ernst Straus, I now have an Einstein number of 4. And an Erdös number of 3. Hurray!
Finally my favourite Einstein story. Einstein and wife (apparently in 1931) toured Hubble’s lab and viewed his telescope. When told that the impressive instruments were what Hubble used to study the shape of the universe, she replied:

Well, my husband does that on the back of an envelope.

As Sheldon Cooper would say:”Bazinga!”

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Under Promise and Over Deliver

Mr Lee Kuan Yew, the man responsible for making modern Singapore what it is today, passed away on 23 March 2015, aged 91. Incidentally it was also Emmy Noether’s birthday. I have wanted to start off my class by showing the google doodle but the lesson plan had to change in view of the more sombre and relevant news. The mathematical exploits and Emmy and Sophie will have to wait for another day.

Throughout this whole week of national mourning, much have been wrote about his sagely advice. In particular, the current minister of defence said that Mr Lee often reminded the younger ministers to “under promise and over deliver.” Something worth remembering in this blow-your-own-trumpet kind of world. There is also much mentioned from down under when he harshly commented in 1980 that if Australia did not shape up, it would become the “poor white trash of Asia.” Talk about not mincing his words. Well, the positive side is Australia as a nation — and the former PM acknowledged it on record — took the advice seriously, reformed and emerged after a quarter of a century to be a strong economic force.

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Kobon Triangles

Students occasionally have great ideas. Was discussing a problem that originated from some students but was not very well posed. We managed to reformulate it as the maximum number of triangles that can be formed with n lines. This turns out to be well known and already discussed by Gardner who stated that the problem came from Kobon Fujimura. A link to a MAA column by Ed Pegg Jr as well as the OEIS entry. The problem is incidentally still not completely solved.

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Translate

It’s slightly old but I only recently saw this article about how Google Translate make use of linear transformation.

The new book on my desk eta products and theta series identities has the following quote in the preface.

In der Theorie der Thetafunctionen ist es leicht, eine beliebig grosse
Menge von Relationen aufzustellen, aber die Schwierigkeit beginnt da,
wo es sich darum handelt, aus diesem Labyrinth von Formeln einen
Ausweg zu finden. Die Besch¨aftigung mit jenen Formelmassen scheint
auf die mathematische Phantasie eine verdorrende Wirkung auszu¨uben
– G. Frobenius, 1893

The quote was rendered as:
In the theory of Thetafunctionen it is easy to an arbitrarily large Establish set of relations, but the difficulty starts here where it is a question of this labyrinth of formulas a
To find a way out. The preoccupation with those formula masses seems auszuuben to the mathematical imagination a searing effect.

Perhaps, I can feed the same passage in a years’ time to see whether the translation has improved.

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Nested radicals

I was trying to solve an olympiad type problem involving a nested radical of the form
[tex] \sqrt{a+b\sqrt{r}}.[/tex]
I had managed to discover that [tex] \sqrt{a^2- b^2r} [/tex] is an integer but it turned out the trick is to rewrite [tex] \sqrt{a+b\sqrt{r}} = c + d\sqrt{r}.[/tex]

Of course, one naturally asks if this is a specific incident or is there a general theory. This lead to digging up an article that I painstakingly photocopied from the library from back when photocopying was the norm. The article in question is by Susan Landau from 1994 in the Math. Intelligencer titled “How to tangle with a nested radical.”

A simplified version of Theorem 1 is this:
Let [tex] k [/tex] be a field extension of the rational numbers and [tex] a, b, r \in k [/tex] but [tex] \sqrt{r} \notin k [/tex]. Then
[tex] \sqrt{a^2- b^2r} \in k [/tex] is equivalent to [tex] \sqrt{a+b\sqrt{r}} \in k(\sqrt{s}, \sqrt{r}) [/tex] for some [tex] 0\ne s \in k [/tex].

For example, one may check that
[tex] \sqrt{5+2\sqrt{6}} =\sqrt{2}+\sqrt{3} [/tex].

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Millenium Bookball

is a pretty cool mathematical sculpture by George Hart. Gone with the Wind, Charlie and the Chocolate Factory, The Cat in the Hat, Green Eggs and Ham. What is not to like? Speaking of which, I still cannot believe that it was only because of my son who is now four that I read Dr Seuss for the first time in my life. Growing up with parents who do not speak English, I only started reading English books in Primary School. I still remember the joy when I did well in school examinations and the prize was to go to the school hall and select one free book for keeps.

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On practice

I’m sure I am interpreting this in a context different from Vygotsky who was quoting Lenin

Man’s practice, repeated a billion times anchors the figure of logic in his consciousness.

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Math videos

“Speaker, I’d like to talk about twin prime numbers …” goes McNerney in the US Congress. This took place on 11 Feb 2014. More details may be found here. There is still hope in politics afterall. I would love to see the day when mathematics is discussed in our parliament.

I’ve longed known about the discussion on the Monty Hall problem in the movie 21. But it was preceded by a short take on Newton’s or Newton-Raphson method. See the clip here at Mathematics in Movies site.

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Prime Mystery

A nice puzzle from Aziz Inan on plus.maths.org

The number N represents the first 6 digits of a special number. N consists of three prime numbers put side by side. These three prime numbers come xth, yth and zth on the list of primes, where x, y and z are themselves three consecutive primes (for example, x, y and z could be 3, 5 and 7, in which case we’d be looking at the third, fifth and seventh prime numbers). In addition, if N is split in the middle into two separate numbers, the prime factors of the left part of N add up to its right part. What is N? And what is the special numbers whose first 6 digits it forms?

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10 surprising things about our brains

an article from huffington. Worth the read.

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“Gods” make Comeback at Toyota

is the headline of this bloomberg piece that reports the strange phenomena that human beings are replacing robots in the manufacturing lines. The point is that by automating some of these production process, much of the skill learnt by the human worker is lost. Without these knowledge, the humans cannot innovate and improve on the current process.

I see a close parallel between this and how students are losing their mathematical/computational ability because they are doing all their calculations on a calculator. Sure, if all the mathematics that you will do in future is to add some numbers on your bills, then by all means do that. But if you hope to develop a more mathematical/algebraic thinking then the hours of doing addition and multiplication sums are not something you should dispense with.

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Come Together

is not the beatles song but the title of episode 16 of a tv program called discover science. I happened to catch it on local tv and the combination of quirky Japanese humour and the clever introduction to mathematics and science caught my attention. A preview of the episode is available at the above link but you need to watch the whole episode to find the link to parabolas.

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Mathematics and poetry

A nice article by Ornes on slate and a paper by Glaz.

My favourites has to be the Fermat Last Theorem Poetry Challenge which I have read about previously elsewhere. Here’s one entry by
E. Howe, H. Lenstra, D. Moulton:
“My butter, garcon, is writ large in!”
a diner was heard to be chargin’.
“I HAD to write there,”
exclaimed waiter Pierre,
“I couldn’t find room in the margarine.”

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Maths & Fashion

That headline caught my eye and earned the newspaper article a more detailed look. It’s about a sub brand from Issey Miyake called 132.5, which apparently stands for 1 piece of material which forms a 3d dress but yet can be folded flat into 2d. The meaning of the .5 wasn’t that clear to me.

Fashion’s not my thing but this is certainly cool enough to merit a link.

http://www.isseymiyake.com/en/brands/132_5.html

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Advice to Young Mathematicians

The Princeton Companion to Mathematics contains a short section of sound advice to young mathematicans by Sir Micheal Atiyah and others. It is certainly worth reading.
The section can be downloaded from here.

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Quadratic reciprocity

– a proof due to Zolotarev and its connection to dealing cards.

A list of 240 and counting published proofs of the theorema aureum.

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Of word problems and place values

If you google “Abbott & Costello 7 x 13 = 28″, you can get a funny sketch of how to show 7 x 13 = 28 by dividing, multiplying and adding. There are a couple of versions, and in one of them, Abbott ask Costello

A: “did you ever go to school stupid?”
C: “Yes sir, and I came out the same way!”

Another classic one goes like this

A captain owns 26 sheep and 10 goats. How old is the captain?

According to the book Making Sense of Word Problems by Verschaffel, Greer and de Corte, children from different parts of the world have actually offered answers to the question.

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The broken stick problem

A slick solution to the broken stick problem or otherwise known as the uncooked spaghetti problem

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That’s an emphatic NO!

all 49% of it.

Seen via math-fail.com
I was curious about the source and google pointed me here.

To restore our sanity, here’s a neat problem from the 2012 AMC 8:
Let R be a set of 9 distinct integers. Six of the elements are 2, 3, 4, 6, 9 and 14. What is the number of possible values of the median of R.

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