# NPTEL: An Innovation in Visual and Online Learning

First of all, Happy Independence Day to all my Indian Friends and followers. This post is about an Indian pioneer in online learning, namely NPTEL.
National Programm on Technology Enhanced Learning (NPTEL) provides E-learning through online Web and Video courses in Engineering, Science and humanities streams. The mission of NPTEL is to enhance the quality of Engineering education in the country by providing free online courseware. All videos of NPTEL include the lectures of Indian professors in IITs and IISc and they can be found either on NPTEL homepage or at their YouTube page. On Youtube, the Video Courses are organised as PLAYLISTS under the following Categories:
1. Core Sciences
2. Civil Engineering
3. Computer Science and Engineering
4. Electrical Engineering
5. Electronics and Communication Engineering
6. Mechanical Engineering

.
And on their website, these are arranged in a order of Subjects.
$textnormal {Visit NPTEL Website}$
$textnormal{Watch Videos On YouTube.com}$
Bellow are two Course Videos, as a demo, one on Semiconductors and other on Artificial Intelligence. It would be better to watch them, before you go for whole.

# A Problem On Several Triangles

A triangle $T$ is divided into smaller triangles such that any two of the smaller triangles either have no point in common, or have a vertex in common, or actually have an edge in common. Thus no two smaller triangles touch along part of an edge of them.
For an illustration let me denote the three vertices of T by 1, 2 and 3. Now number each of the vertices of the small triangles by 1, 2, 3. Do this in an arbitrary way, but such that vertices lying on an edge of T must not be numbered by the same number as the vertex of T opposite to that edge.

Show that among the small triangles there is always one whose vertices are numbered by 1, 2 and 3.

# Solution

To show that among the small triangles there is always one whose vertices are numbered by 1, 2 and 3, we show that the number of small triangles whose vertices are labeled with $1,2,3$ is odd and thus actually $>0$ !

We enumerate all small triangles in the picture as $T_1$ , $T_2, \ldots, T_n$ and denote by $a_i$ the number of edges with endpoints $1$ and $2$ in each triangle $T_i$ . Thus, if say the vertices of $T_i$ are labeled by $1,1,2$ , then $a_i=2$ , and so on …

Observe now that obviously we have

$\displaystyle a_1+a_2+a_3+\cdots +a_n= A+2B,$

where $A$ is the number of triangles whose vertices are labeled $1,2,3$ , while $B$ is the number of those triangles labeled by $1,1,2$ or $1,2,2$ . (Actually it is easily seen that $a_i=2$ for such triangles, while $a_i=1$ if the vertices of $T_i$ are $1,2,3$ and $a_i=0$ otherwise.) All we have to show is that $A$ is odd.

Let $C$ denote the number of $12$ -edges lying inside the original triangle $T$ and let $D$ be the number of $12$ -edges lying on the boundary of $T$ . Every interior $12$ -edge lies in two triangles $T_i$ and thus it is counted twice in the sum $a_1+a_2+a_3+\cdots +a_n$ , while every boundary $12$ -edge is counted only once. In conclusion we get

$\displaystyle a_1+a_2+a_3+\cdots +a_n= 2C+D,$

which yields

$\displaystyle A+2B=2C+D.$

Hence $A$ is odd if and only if $D$ is odd. It is therefore enough to show that $D$ is odd.

According to the hypothesis of the problem, edges labeled $12$ or $21$ can occur only on the $12$ -edge of the large triangle $T$ . We start walking along the edge $12$ of the triangle $T$ starting at the vertex $1$ toward the vertex $2$ . Now, only when we first pass an edge labeled $12$ will we arrive at the first vertex labeled $2$ . A number of vertices labeled $2$ may now follow, and only after we have passed a segment $21$ do we reach a label $1$ , and so on. Thus after an odd number of segments $12$ or $21$ we arrive at vertices labeled $2$ , and after an even number of such segments we arrive at vertices labeled $1$ . Since the last vertex we will reach is the vertex $2$ of the big triangle $T$ , it follows that the total number of segments $12$ or $21$ lying on the side $12$ of the big triangle $T$ must be odd! The same reasoning applies for each of the other edges of the big triangle $T$ , so we deduce that $D$ , the total number of $12$ or $21$ -edges lying on the boundary of $T$ , must be odd. Proved

# Graphical Proof

It is obvious. As a result of this numbering we get following diagram:

Problem Image

# Two Interesting Math Problems

## Problem1: Smallest Autobiographical Number:

A number with ten digits or less is called autobiographical if its first digit (from the left) indicates the number of zeros it contains,the second digit the number of ones, third digit number of twos and so on.

Image

# Blog of the Month -August 2011

I announced that I shall chose a blog from the education blogsphere as Blog of the Month. To complete this task, I googled for days, read them, analysed them and now I have the winner of ‘Blog of the Month’.
This is the first month of this series and discussing article is made in hurry, so one can feel an emptiness and lack of interest in it. But believe, Blog of the monthwas not selected in hurry. I took quick looks on about 500 blogs and thousands of posts. I created a list of all blogs I read and rated them on behalf of their qualities, visitors, content, language etc. From the list of 513 blogs, the shortlisted blogs were:

1. What’s New (math)
2. Gödel’s Lost Letter and P=NP(Math and Computer Science)
3. Peter Cameron’s Blog(math)
4. Let’s Play Math(math)
5. Unapologetic Mathematician(math)
6. Cock Tail Party Physics(Physics)
7. WordPress Tips(Blogging)
8. Honglang Wang’s Blog (Math and Programming)
9. The GeomBlog(CS)
10. Republic Of Mathematics (Math and Media)

I count a lot of things that there’s no need to count. Just because that’s the way I am. But I count all the things that need to be counted.

And Yes! The blog of the month is Peter Cameron’s Blog with useful content, interactive language and multidimensional approach to mathematics.

Peter Cameron is a professor of mathematics in London and he writes about math, media and education at http://cameroncounts.wordpress.com. He mingles everything with math, like poetry – media – fun and internet. His blog is full of Expositories, Problems and Results, Posts about doing – playing and learning mathematics, Poetry, Events Talks and Conferences, Typesettings and Mathematics in Media. A list of categorized posts can be found here.

# Reviews

Rating: 8.9/10
View: 7.0/10
Content: 9.5/10
Interaction: 9.0/10
Language: 9.5/10
Frequency of Posts: 8.5/10
Content Management: 10/10

[Last Updated: 20:03 IST 2011/08/05]

What are you views and thoughts on this selection? Rate Peter Cameron’s blog on the base of 10. Your comments are heartly welcomed.

# Chess Problems

1. In how many ways can two queens, two rooks, one white bishop, one black bishop, and a knight be placed on a standard $8 \times 8$ chessboard so that every position on the board is under attack by at least one piece?
Note: The color of a bishop refers to the color of the square on which it sits, not to the color of the piece.
2. Can you attack every position on the board with fewer than seven pieces?

# Solution

1. Two ways as follow: