MY DIGITAL NOTEBOOK

A Personal Blog On Mathematical Sciences and Technology

Home » 2011 » February

Monthly Archives: February 2011

On World Math Day – Do A Simple Self-Test to Identify a Creative Mathematician in You

Monday, February 28th, 2011 22:01 / 1 Comment

Queen Christina of Sweden (left) and René Descartes--- Image via Wikipedia

There are many mathematicians, chemists, musicians, painters & biologists among us. But they are unaware of their qualities. Only one small suggestive strokes are needed for them. Unless one tells then, they don’t know how great they are.
(more…)

The Collatz Conjecture : Unsolved but Useless

Saturday, February 26th, 2011 21:59 / Leave a Comment

The Collatz Conjecture is one of the Unsolved problems in mathematics, specially in Number Theory. The Collatz Conjecture is also termed as 3n+1 conjecture, Ulam Conjecture, Kakutani’s Problem, Thwaites Conjecture, Hasse’s Algorithm, Syracuse Problem.

(more…)

Six Puzzles

Friday, February 25th, 2011 08:25 / 2 Comments

Assume that the English letters are digits (from 0 to 9 ) and they satisfy the given relations, then you have to solve each equation for these letters.

For Example:
$ABCDE \times ABCDE = FDBABCDE$ can have a solution:
$09376 \times 09376 = 87909376$

Similarly, Try these:

1. $ABCDEEABCD$ $\times$ $FEC$ = $AAAAAAAAAAAA$

2. $(A+B+C+D+E)$ $\times$ $(A+B+C+D+E)$ $\times$ $(A+B+C+D+E)$ = $ABCDE$

3. $6 \times ABCDEF =DEFABC$

4. $ABCDEABCDEABCDE$ = $C \times CCCCCGGGGH \times ABCDE$

5. $ABCDEABCDE$ $= FF \times GHGF \times 86485$

Solution:

1. $8547008547 \times 104$ $=888888888888$
2. $(1+9+6+8+3) \times (1+9+6+8+3) \times (1+9+6+8+3)$ $=19683$
3. $6 \times 14857$ $857142$
4. $283512835128351$ $=3 \times 3333366667 \times 28351$
5. $8648586485$ $=11 \times 9091 \times 86485$

Puzzle Idea: Mr. Sawinder Singh, Gurdaspur, Punjab (INDIA)
Note:
There may be many other solutions for these puzzles too.

Dirichlet’s Theorem and Liouville’s Extension of Dirichlet’s Theorem

Thursday, February 24th, 2011 17:30 / Leave a Comment

Topic

Beta & Gamma functions

Statement

$\int \, \int \, \int_{V} \, x^{l-1} y^{m-1} z^{n-1} dx \, dy \,dz = \frac { \Gamma {(l)} \Gamma {(m)} \Gamma {(n)} }{ \Gamma{(l+m+n+1)} }$
where V is the region given by $x \ge 0, y \ge 0, z \ge 0, \, x+y+z \le 1$ .

(more…)

The Lindemann Theory of Unimolecular Reactions

Tuesday, February 22nd, 2011 14:40 / 11 Comments

It is easy to understand a bimolecular reaction on the basis of collision theory.

When two molecules A and B collide, their relative kinetic energy exceeds the threshold energy with the result that the collision results in the breaking of comes and the formation of new bonds.

But how can one account for a unimolecular reaction? If we assume that in such a reaction $A \longrightarrow P$ , the molecule A acquires the necessary activation energy for colliding with another molecule, then the reaction should obey second-order kinetics and not the first-order kinetics which is actually observed in several unimolecular gaseous reactions. A satisfactory theory of these reactions was proposed by F. A. Lindemann in 1922. (more…)

The Mystery of the Missing Money – A classical puzzle by Human Computer Shakuntala Devi With Solution

Monday, February 21st, 2011 09:35 / Leave a Comment

Puzzle

Two women were selling marbles in the market place — one at three for a Rupee and other at two for a Rupee. One day both of then were obliged to return home when each had thirty marbles unsold. They put together the two lots of marbles and handing them over to a friend asked her to sell then at five for 2 Rupees. According to their calculation, after all, 3 for one Rupee and 2 for one Rupee was exactly same as 5 for two Rupees.
Now they were expecting to get 25 Rupees for the marbles, (10 Rupees to first and 15 Rupees to second), as they would have got, if sold separately. But much to their surprise they got only 24 Rupees ( $60 \times \frac {2} {5}$ ) times for the entire lot.

Now where did the one Rupee go? CAN YOU EXPLAIN THE MYSTERY?

Solution

There isn’t really any mystery, because the explanation is simple. While the two ways of selling are only identical, when the number of marbles role at three for a Rupee and two for a Rupee is in the proportion of three by two. Therefore, if the first woman had handed over 36 marbles and the second woman 24, they would have fetched 24 Rupee, immaterial of, whether they sold separately or at five for 2 Rupee. But if they had the same number of marbles which led to loss of 1 Rupee when role together, in every 60 marbles. So, if they had 60 each, there would be a loss of 2 Rupee and if there were 90 each (180 altogether) they would lose 3 Rupees and so on.
In the case of 60, the missing 1 Rupee arises from the fact that the 3 marbles per Rupee woman gains 2 Rupees and the 2 marbles per Rupee woman loses 3 Rupees.
The first woman recieves 9½ Rupees and the second woman 14½ , so that each loses ½ Rupees in the transaction.

The Fear, and the Use, of Mathematics and Physics

Friday, February 18th, 2011 23:10 / 3 Comments

Image via Wikipedia

The two areas of human enquiry that inspire the greatest terror in the hearts of students are undoubtedly mathematics and physics. You may find history or chemistry or economics difficult, but your reaction to these subjects, and more others is almost certainly not fear. On the other hand, when you encounter an equation, your first reaction is to escape to more amiable company. If you compare subjects to people, you will realize that your reaction to maths or physics is very similar to your reaction to a stern, quiet person who is famous for his wisdom but who makes you very uncomfortable indeed. When he speaks you listen dutifully, because you’ve been told his words contain a lot of meaning, but you understand almost nothing, and you end up feeling foolish and exposed; and what is worse, this person does not need to shout to make you feel this way — he just has to look at you. When you see an equation or a mathematical expression you react in the same way. Let us try to under stand that what mathematics is and why it is so difficult.