Open Day 2006

The Open Day is an annual event in which some selected school students from some selected schools come to visit TIFR for a day. They are taken around laboratories and are given an exposure on how science is done.

We also had put up a Mathematics laboratory (whatever that means). It was more of a puzzles and games laboratory, where the puzzles and games were based on mathematical facts. For example, Shameek did experiments with geometry by introducing the school students to the Möbius band. Arati explained to them the mathematics behind the shape of the bubbles. They learnt about the types of tilings from Anupam. Sagar introduced them to Sperner's lemma. They realized that their hobbies can also have deeper meanings, when Mimansa told them about the mathematical aspect of Origami. Somnath showed them how to use mathematics to solve the problem popularly known as Towers of Hanoi. Four color theorem was done by Shane.

I decided to do something related to probability. I did Buffon's needle problem. I had wrote a flash program to simulate the experiment and used a laptop to demonstrate how one can estimate π by throwing needles on the floor. Those who want to know more about Buffon's needle problem, can visit my short write up here.

I have made a web-page on the small application I had written. You can find it here. You need Flash Player 8 or above to view it properly.

Funny incidents while teaching maths!

I heard some jokes on teaching mathematics. They may be funny but the sad part is that they are not completely away from reality. It is actually quite a sad situation!

Well here are the stories!

Division of numbers and division of money

Some experts wanted to assess the quality of mathematics teaching in remote villages. They went to one such village, and asked the ninth standard students there to divide 1224 by 6. Most of them answered 24. Yeah! That's right, the students answered 24. The people were shocked. After a whole day of discussions, one of the experts thought of testing their knowledge by asking the following question:

Suppose you have got Rs. 1224 and you have to divide it equally among six people. How much would each person get?

This time the students argued,

The total amount of money is more than 1200. So each person will get at least 200. And the extra is 24 which when divided by six is 4. So each person will get Rs 204!

Laws of cancellation

Sandeep (fictional name) teaches pre-calculus in a class. In one of the assignments he asks the following question:The student answered 1. And gave the following explanation:

You cancel off 'x'. So you only have 'sin' left. Now you can take any value of sin. In particular, sin 90 = 1.

Sandeep of course did not give any points for that problem. The student came to ask why he has deducted marks when the answer is correct. And the student got a calculus book to show that the answer is correct. Then Sandeep tried to explain that the proof the student gave was not logical. The student asked,

Why does everything have to be logical?

Cube roots, Square roots and hands
One student went to his TA saying that she cannot solve "³√x = 2". The TA started to explain.

TA : You are given cube root of x is 2. You have to remove the cube root sign on the left. What would you do?
Student: I'll take cube roots on both the side.
TA: But you already have a cube root.
Student: I do not understand. Still I can take cube roots.
TA:(flabbergasted) Okay, let us first solve an easier problem: How would you solve, "√x = 2"?
Student: I'll take square root on both sides.
TA: But you already have square root on the left.
Student: Oh now I understand. I'll take cube root on both sides.
TA:(almost throwing in the towel) I think you are not getting the point. Let me take up a real life situation. Suppose you are wearing gloves and I want to see your nails, what would you do? Would you wear another pair of gloves?
Student: HOW DARE YOU WANT TO SEE MY NAILS! WHY SHOULD I SHOW MY NAILS TO YOU? (Gets angry and leaves the office)

What do I want to do with this blog?

For quite sometime I have been thinking writing something in my blog, but for a variety of reasons I simply could not do it. Well, first of all, being towards the end of my Ph.D. program, I am quite busy tying up the loose ends. Further, the nature of this blog itself is a bit too daring. I intend to expand on the last statement a bit.

Till now I was writing about things somehow related to mathematics. I tried to keep the stuff simple, still interesting. Somehow suddenly I started feeling a bit of a shortage of simple things to write in the blog. Most of the things are already there on the net; and most likely you do not have to go anywhere further than Wikipedia. Then I wonder, why should I waste time writing on things which are already well documented on the net. And specially when the other people are doing a good job, definitely a much better job than what I could have done alone.

For example, for the last few months I had been thinking for writing a blog on projective planes. I read a bit and then thought, "Okay, I need more interesting things if I have to write a popular article." I went on reading and then slowly it started getting technical. And by the time it got to some level of interest, I already was dwelling in open problems in the area. Then I decided to write a part of it. Before writing, I did a google search on those topics. There were so many articles dealing with exactly what I wanted to write about, I felt it is better that I do something else which will be more original.

Thus I decided to change the flavor of the blog.

This blog I'll keep to update about things happening around me. Of course, only those of academic interest. I may try my hand at writing about some nonacademic incidents also. But

Warning: I am very bad at narration. So read at your own risk! :-).

Blogs on Maths

Sorry for the delay in this blog. I'm a bit busy nowadays and that has affected the rate at which I blog quite drastically. Well I think now for a few months my blogging rate might be varying a lot. But I hope there will be at least one update per month.

I dedicate this blog to other blogs. One of my blog readers was asking me to start a Math blog with applications in mind. That induced me to search the net for Maths blogs. There shouldn't be much of an overlap, you see. Well the number of blogs on Maths seem to be quite impressive. Of course, I don't have the time and energy to go through all of them. But I did find a few which I found interesting. Some are easy to read. Some are technical. Here are a few:

A tribe

I found some of the discussions here quite good. Nice online community on Maths! :)

Some school level maths

There are some sites which remind you of the school days. Here is a site which school students will definitely like. It gives a flavour that Maths is actually a lot of fun.

Homeschool Math

This is similar to the above but the blogs deal with slightly higher topics.

Good Math, Bad Math

Thanks to Samudrika for the link to this blog! This has a bit of a computer science flavour in it.

Namespace collision

This blog does maths in a rigorous fashion in the mathematician's way. Its Maths postings tend to be a bit technical. Well it is still for beginners. The next one is completely technical.

Interesting Mathematics

Well I was wondering if there are blogs like this for every sub branch of Mathematics! No idea still now. Well I'm now tired of searching! :). The next one will be a lighter blog.

Shripad's Math fun facts!

Nice facts!

And thus ends my tour of blogs. There are thousands of other nice ones I'm sure! I'll write on them as and when I find them.

Well I'll write something original soon. My wife's coming home on her summer break next month. Just busy trying to get myself to be as free as possible during her visit here! So I will be very irregular in posting for about a month from now. (And may be quite irregular till August!) :)

But do let me know if you land up on some interesting sites!

Hash Tables

Couple of days back, I got into a discussion about hash tables with one of my friends. Initially we were confused about what good it is over an array. Before explaining the confusion, I'll describe an example. (You know I strongly believe what my sixth standard teacher told me once: "Eg makes it easy.")

Suppose we want to store a table containing roll numbers and names of students:
(1, Pundarikakshya Purakayastha)
(2, Prasenjit Sarvadhikari)
(... well I think it is better to shorten the names ...)
(3, Arun)
(4, Varun)
(5, Darth Vader)

Okay we can continue with this. Well we can store this data in an array containing five elements and store the ith entry in a[i-1]. That would be good enough. We were confused what is the point of defining something like a hash table. Well we realised the point is we don't construct hash tables for such data. To give a real example, hash tables are used to handle data like path names of Linux executables indexed by the command names, like (just as an example)
(ls, /bin/ls)
(rm, /bin/rm)
(shutdown, /sbin/shutdown)
(vi, /usr/bin/vi)
(acroread, /usr/local/bin/acroread)
... and so on

Suppose there are 1355 commands and if we put the whole data in a 2 x 1355 array, it might be a time consuming job to find the path for a command you want. First you have to find the position of the command in the array and then check the corresponding path. If the command lies towards the end of the array, it will take some time to find it out.

So the way a hash table solves the problem is that it uses a hash function to convert the command name into a number and then store the path in a one dimensional array at the corresponding place. Like if, for example, for a suitable hash function, "ls" converts to 6, the computer sets a[5] = "/bin/ls". And so on. So when the computer has to retrieve the path for ls, it calculates the value of the hash function at "ls" and gets 6 as it's value and then gets the path at a[5] to be "/bin/ls". That works much faster than searching an array for "ls".

Now, as usual, life is not all that fair and it's very hard to construct hash functions which are "good" in the sense that they do not give the same number for two different commands and at the same time the numbers it produces are not too huge compared to the total number of commands. Well if one is interested how one solves those problems, one can read the article on Hash Tables in Wikipedia or some advanced book on algorithms.

Memories from my mathematical childhood

The reason for writing such a strange blog is the sudden realisation that sometimes we don't give enough importance to the things we have around us. I don't want to get too philosophical but all this started when I was looking for a book which I treasured a lot in my schooldays. My father is a great admirer of Mathematics and used to get me books on Mathematics and mathematical puzzles. I used to like them. One of them which was my favourite at that time was by Ya. I. Perelman. It was titled "Mathematics can be Fun". There was another book by him: "Physics can be Fun" which also I liked. Now I don't know where they are. I just searched Google for the book. Got an Amazon link. "Mathematics can be Fun" is out of print.

I spent a lot of time with the book. I specially remember and liked the discussion on Diophantine equations and pythagorean triples. I was fascinated by the fact that, for three integers a,b and c, whenever a² + b² = c², there exists integers m and n such that

a = m² - n²,
b = 2mn,
c = m² + n².

The fact that one can logically argue to get these forms really fascinated me in my high school. I also liked the section on maxima and minima done with just some basic knowledge in high school algebra.

Anyway, I really liked the book and it's really sad that Mir had to stop publishing.

My first blog

I had been thinking of blogging for quite some time. The reason for blogging is completely personal. I'd like to put up some random stuff so that I don't lose them and so that I can share them with others more easily. Well the stuff will be mostly basic math trivia and computer related stuff though I don't really want to restrict myself about what I want to put in this blog right now. For example I might put up a few poems too. Or movie reviews. Anything. But then this is meant to be a serious blog and will be boring to most people.

Read at your own risk! :).

To the few friends who were asking me to blog for quite some time: Well finally, I'm here!