Solve it - Question 4

Dear All
Try this question:

Each of the integers 226 and 318 have digits whose product is 24. How many three digit positive integers have digits whose product is 24?

Quite simple...
Check your answer here.

Solve it - Question 3

Dear All

Solve it!
Click here for answer.

Solve it - Question 2

Dear All

The given figure is formed by two squares. The side of each square is a whole number. If area of the figure is 58 cm^2, find its perimeter.
Click here to check your solution.

Solve it - Question 1

Dear All

Find the sum of digits of number 1000^20 – 20 expressed in decimal notation.



Click here for answer.

A view of Mathsland

---

DEAR ALL

To most people, mathematics is that subject they have always had difficulty understanding. It is a form of communication, a kind of strange language in which complete sentences must have something called an equals sign or some other equally strange symbol. It appears to be a form of the English language but interlaced with rows of austere symbols and incomprehensible formulae (some Martian language!).

For different reasons, the majority of the world's `educated' population, by the time they graduate from high school, have already made up their minds that mathematics is difficult and that nothing new ever happens in mathematics. My suspicion is that this uninformed majority spreading these unfounded rumors have no personal experience with mathematics. They probably heard this story from a friend who in turn had heard rumors from elder brothers and sisters that mathematics is a difficult subject. Believing this lie and obviously lacking self-confidence and motivation, most decide to give up before they even give it a try.

The world of mathematics is an ever-changing one. It is a world highly endowed with provocative ideas, very rich in poetry and full of vivid images.

To read the complete article, click here.

'True' life story - Nobert Weiner

---

Dear All

Weiner, a great mathematician of this century, was very absent minded. The following story is told about him: When he and his family were moving house, his wife was so certain that he would forget that they had moved and where they had moved to that she wrote down the new address on a piece of paper, and gave it to him. Naturally, in the course of the day, an insight occurred to him. He reached in his pocket, found a piece of paper on which he furiously scribbled some notes, thought it over, decided there was a fallacy in his idea, and threw the piece of paper away.
At the end of the day he went home --- to his old address. When he got there he realized that they had moved, that he had no idea where they had moved to, and that the piece of paper with the address was long gone. Fortunately inspiration struck. There was a young girl on the street and he conceived the idea of asking her where he had moved to, saying, "Excuse me, perhaps you know me. I'm Norbert Weiner and we've just moved. Would you know where we've moved to?" To which the young girl replied, "Yes daddy, mommy thought you would forget."

Why is there no Nobel Prize in Mathematics?

---

Dear All

Six Nobel Prizes are awarded each year, one in each of the following categories: literature, physics, chemistry, peace, economics and medicine. Notably absent from this list is an award for Mathematics. The reason for this conspicuous omission has been subject of extensive speculations, some of which are discussed below.

One of the most common reasons as to why Nobel decided against a Nobel prize in math is that [a woman he proposed to/his wife/his mistress] [rejected him beacuse of/cheated him with] a famous mathematician. Gosta Mittag-Leffler is often claimed to be the guilty party.

There is no historical evidence to support the story.

For one, Mr. Nobel was never married.

Click here to read the complete article.

The prof's age

Dear All
Its just for fun...

Professsor: Given the age of the light as 3 x 10^8 m/s and the density of water as 1km/m^3, what's my age?

Student: I'd say about -- 46.

Professor: Excellent! How did you know that?

Student: You see, sir, my brother's 23 and he's only half-mad.

Train alert!

A group of people were asked the following question. ``Say there are 2 people tied to a railway track, and a train is fast approaching. You have time to save just one of them. Which one would you save?''

The politician replied ``After arranging for a television crew to be present, and preferably making sure there was a press conference afterwards, I would rescue the one with the louder screams.''

The accountant replied ``I would rescue the one who offered to pay me most.''

The journalist replied ``I would ask the train driver to delay the collision till I could get a camera.''

The lawyer replied ``I would jump into the train and offer my services to the driver, who will almost certainly be sued by relatives of the two. ''

The physicist replied ``I would derail the train, as your question did not place any limitations on the safety of those in the train.''

The statistician replied ``I would toss a coin to pick one of them. Then I would toss the coin another hundred times to make sure that that was not a statistical fluke. Then I'd rescue the one selected.''

The Mathematician replied ``I would make two rescues, each saving half a person. But since persons come in whole numbers, in the mathematics of persons 1/2 is always rounded to 1, so each rescue saves a person. There are two rescues, so 2 people are saved.''

Measuring the height of trees

Dear All

Some Native Americans had a very interesting way of doing this. To see how high a tree was, they would find a spot where, looking under their legs (as shown), they could just see the top of the tree. The distance from such a spot to the base of the tree was approximately the height of the tree.

Why does this work? The reason is quite simple. For a normal, healthy adult, the angle formed by looking under one's legs is approximately 45^o. Hence, the distance to the tree must be around the same as the height of the tree.

Poor Man's space travel

I met a youngster rummaging through a dust bin. He seemed to be interested only in large sheets of paper.
"What are you doing?"
"I am trying to get to the moon."
"Are you going to make a paper spaceship?"
"No, it's much simpler than that."
He put one piece of newspaper down and stood on it.
"I am now nearer to the moon."
He doubled the paper and stood on that. Then doubled again and stood on top of that -- there were now 4 thicknesses of paper, say a total of 4/10 of a millimetre and he carried on doubling.
After a few more doublings I began to get the idea. It is roughly 400000km to the moon. How many times must he double?
Surprisingly, the answer is only 43.
The pattern is 1, 2, 4, 8, 16,..., each term doubling the previous one. Such a sequence is called a Geometric Progression and the nth term is given by 2^(n-1).
Geometric Progressions (GP's) often have terms which get very big like this one. For some GP's however, the terms get smaller, look at the series1, 1/2, 1/4, 1/8, 1/16,... for example.
The population explosion is often analysed by considering population increase, e.g. 3% as being added at the end of a year. The GP is made by multiplying by 1.03 to get successive terms.
``Population increases geometrically: Food increases arithmetically. Population will therefore always outstrip food supply until famine, disease or war ensue.''

The most touching story in Mathematics

``And if you divide any number by itself, you get 1.''

The teacher in a small high school in southern India turned round to see a tiny hand trying to reach the ceiling. Oh by the gods, him again! That Aiyangar boy with his horribly difficult and quite irrelevant questions. Like last week, when he wanted to know how long it would take for a steam train to reach Alpha Centauri. As if he would be able to afford the fare if he knew. Well, he couldn't let him exercise his hands too much.

``Yes Ramanujan?''

The small boy with shining eyes stood up. He spoke slowly, with the calm confidence of one who did not need to be told he was the best in the class.

``Is zero divided by zero also equal to one?''

Unfortunately for all those other teachers who've been asked this question at least twenty times in their lives, the response to the question is unknown. But the life of the boy, Srinivasa Ramanujan Aiyangar, certainly isn't.

To read this complete article, click here.

Fermat's Last Theorem

Dear All

In the margin of his copy of a book by Diophantus, Pierre de Fermat wrote that it is possible to have a square be the sum of two squares, but that a cube can not be the sum of two cubes, nor a fourth power be a sum of two fourth powers, and so on. Further, he wrote that he had found a truly marvelous proof which the margin was too small to contain.

Fermat's Last Theorem states that xⁿ + yⁿ = zⁿ has no non-zero integer solutions for x, y and z when n > 2.

That is to say, there are no integers x, y, z such that x³ + y³ = z³ or integers x, y, z such that x⁷ + y⁷ = z⁷.

Although this is easily stated, it has proved to be one of the most puzzling problems in the whole history of mathematics. Long after all the other statements made by Fermat had been either proved or disproved, this remained; hence it is called Fermat's Last Theorem (actually, Conjecture would be more accurate than Theorem).

This conjecture was worked on by many famous mathematicians. Fermat himself proved this theorem for n = 4, and Leonhard Euler did n = 3. Special cases were dispatched one after another. New theories were developed to attack the problem, but all attempts at a general proof failed. They failed, that is, until this decade, when, building on work of many famous mathematicians, Prof. Andrew Wiles of Princeton University finally proved it in 1996. His method could not have been known to Fermat. Fermat's "truly marvelous proof" is now believed to have been faulty.

The actual proof is very indirect, and involves two branches of mathematics which at face value appear to have nothing to do either with each other or with Fermat's theorem. The two subjects are elliptic curves and modular forms and involve work done previously by Taniyama and Shimura. The greatest difficulty was in proving that the Taniyama-Shimura conjecture was true. This is the contribution made by Andrew Wiles, and the final stage in establishing Fermat's Last theorem.

To know more, click here.

Number of Tangents

Dear All

Today a student asked me a question: How many tangents can we draw at a point on a sphere? We are well aware that to a point on a circle, the answer is ONE.
You may imagine a football (sphere) lying on the floor (tangent plane).
What are your views? Do share.

The equations of knowledge and power

Here's one of those things friends send to friends over the Internet, leaving the original author lost (or simply hiding).

After applying some simple algebra to some trite phrases and cliches a new understanding can be reached of the secret to wealth and success. Here it goes.

         Knowledge is Power,
       Time is Money, and (as every physics student knows)
       Power is Work over Time.

So, substituting algebraic equations for these time-worn bits of wisdom, we get:

             K = P   ...(1)
           T = M  ...(2)
           P = W/T  ...(3)

Now for a few simple substitutions. Put W/T in for P in equation (1), which yields:

             K = W/T  ...(4)

Put M in for T into equation (4), which yields:

             K = W/M  ...(5)

Now we've got something. Expanding back into English, we get:

Knowledge equals Work over Money.

What this MEANS is that:

Theorem 1. The More You Know, the More Work You Do, and

Theorem 2. The More You Know, the Less Money You Make.

Solving for Money, we get:

            M = W/K  ...(6)

Money equals Work Over Knowledge.

From equation (6) we see that Money approaches infinity as Knowledge approaches 0, regardless of the Work done.

What THIS MEANS is:

Theorem 3. The More you Make, the Less you Know.

Solving for Work, we get

         W = MK  ...(7)

Work equals Money times Knowledge

From equation (7) we see that Work approaches 0 as Knowledge approaches 0.

What THIS MEANS is:

Theorem 4. The rich do little or no work.

(Don't take it seriously! Read the label for this post.)

Palindrome Number

Hello All

A positive integer N is a Palindrome if the number obtained by reversing the sequence of the digits of N is equal to N. The year 1991 was the only year of the last century which was a palindromic year.
Read this document to solve this interesting problem.
Write your answers in the comment section.

Other Softwares

Hello All

I wish to share with you all some good software's (not related to Maths). Hope you will like them.

1. Portable Dictionary: Dictionary is a encyclopedic dictionary that can be used with or without a screen reader. The dictionary is based on the Wordnet 2.1 database and contains over 250,000 words. These include historical figures, slang and jargon. Click here to download it.
   

New Year came a second late...

Hello All
I know its late to share this piece of information with you... but yet its worth reading!

True. It is indeed a fact. Incredible, unbelievable but that is the real! Anyone who cared to observe keenly would have noticed that on 1 January 2009 at sharp 5:30 a.m. IST one extra second appeared! After 5:29:59 a.m. we had 5:29:60 a.m. and then 5:30:00 a.m. In normal circumstances 5:29:59 a.m. should be followed by 5:30:00 a.m. There is no time called 5:30:60 at all! Such addition of a second is called ‘leap second’, very similar to addition of a day in the leap year. This was done because; sharp at 5:30:00 a.m. it was 24:00:00 at Greenwich Mean Time. In the clock at Greenwich, 31 December 23:59:59 p.m. was followed by 23:59:60 and then moved on to 0:00:00. So at precisely 23:59:60 at Greenwich, England, on New Year’s Eve, there was a one-second void before the onset of midnight and the start of the New Year.

To read the complete article, click here.

Two Exams Taken By Ramanujan

Dear All

On Thursday and Friday, December 3 and 4, of 1903, Srinivasa Ramanujan, who was to become the greatest Indian mathematician in his country’s history, sat for the Matriculation Examination of Madras University.
Click here to read the exams given by Ramanujan.
Try to solve it! Well... Maths is really simple NOW!

Math Assignments For You

Dear Students

I am very glad to make available all the assignments already given to you in the school (Class X, XI and XII). I will also upload all the assignments that you will be getting in near future.
Do practice and enjoy Mathematics!
Follow this blog and be updated...

CLASS IX

MY BOOK: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Challenging Olympiad

CLASS X

Real Numbers
Polynomials
Linear Equations Word Problems
Quadratic Equations
Arithmetic Progression
Triangles
Introduction to Trigonometry
Applications of Trigonometry
Co-ordinate Geometry
Circles
Area Related to Circles
Surface Areas and Volumes
Statistics
Probability

MY BOOK: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Theorems Challenging Olympiad

CLASS XI

Enrichment Assignment (Not from examination point of view)
Sets
Relations and Functions
Trigonometry

Mathematical Induction & Linear inequations

Complex Numbers & Quadratic Equations
Sequence and Series
Limits
Derivatives
Straight Lines
Conic Section
Permutation and Combination

Bionomial Theorem
Probability

CLASS XII

Relations and Functions
Inverse Trigonometry
Matrices
Determinants
Continuity and Differentiability
Differentiation
Applications of Derivatives

Assignment based on all topics (BOOK 1)

Enrichment Assignment -I (Not from examination point of view) Solutions

Indefinite Integrals      Enrichment Assignment (Not from examination point of view)
Definite Integrals
Applications of the Integrals
Differential Equations
Vector Algebra
Three Dimensional Geometry
Probability

Linear Programming

Assignment based on all topics (BOOK 2)

Zeno's Paradox

In the fifth century BC the Greek philosopher Zeno of Elea posed 4 problems, now known as Zeno paradoxes that were intended to challenge some of the ideas concerning space and time held in his day. Zeno’s second paradox concerns a race between the Greek hero Achilles and a Tortoise that has been given a head start. He said 'The slower when running will never be overtaken by the quicker; for that which is pursuing must first reach the point from which that which is fleeing started, so that the slower must necessarily always be some distance ahead.'

Zeno argued, as follows below, that Achilles could never pass the tortoise. Given that Achilles is the fastest Greek hero, Tortoise is one of the slowest animals, the speed difference between is very big, so that the conclusion that Achilles can never pass the Tortoise is contradictory with everyday experience.

Zeno argues as follows: Suppose that Achilles starts at position a₁ and the Tortoise starts with the position t₁ having a head start in terms of the distance. When Achilles reaches the point a₂ = t₁, the Tortoise is already further away at position t₂. When Achilles reaches the point a₃ = t₂,

the Tortoise is at position t₃. This process continues without end, and Tortoise is always ahead of Achilles. The conclusion is that Achilles can never overtake the Tortoise.

Click here to understand the problem in better way!

Here is the simple explanation:

Let us imagine that the race is 25 miles long with a check point at every 5 miles, so that a₁ to a₂ = 5 miles., h₁ to h₂ = 5 miles, and so on. Let’s also imagine Achilles runs 10 mph and the Tortoise runs 5 mph, since Achilles is faster.

Knowing this, we can determine that it will take Achilles 30 minutes to run between each point… Therefore, 30 minutes into the race, Achilles would be at point a₂ and the Tortoise would be between h₁ and point h₂.

An hour into the race, Achilles would be at point a3 and the Tortoise would be at point h₂. Note a₃ = h₂. An hour and a half into the race, Achilles would have passed the Tortoise and be at point a₄, while the Tortoise is between the point h₂ and the point h₃.. Though Zeno’s conclusion that the quicker will never pass the slower (if given a head start) may have been valid during the fifth century B.C., it does not stand true to today’s experiences.

Here is the justification for the fallacy in Zeno's argument:

Metaphysical solution: It's about TIME! Time is not only a physical parameter, but a metaphysical notion as well. Zeno's only take into account Space. Time is completely ignored. In truth, Space and Time are inseparable elements of Cosmos (the Universe). That fundamental truth is valid both physically and metaphysically. Achilles will cover one unit of space in less time than the tortoise. Equivalently, Achilles will cover a longer distance than the tortoise in the same time. By the time Achilles reaches the starting point of the tortoise, the tortoise would have moved a shorter distance. The distance could be so short that Achilles could surpass it in a very short time. Generalizing, the gap the gap reverses. The faster competitor surpasses the slower competitor who had an early start.

Nine Point Circle

Draw a triangle, any triangle (although it may be best to start with an acute triangle).

1) Mark the midpoints of each side (3 points). See Figure 1.

2) Drop an altitude from each vertex to the opposite side, and mark the points where the altitudes intersect the opposite side. (If the triangle is obtuse, an altitude will be outside the triangle, so extend the opposite side until it intersects.) See Figure 2.

3) Notice that the altitudes intersect at a common point. Mark the midpoint between each vertex and this common point. See Figure 3.
No matter what triangle you start with, these nine points all lie on a perfect circle!

Even simple geometry still has some surprises in store! This result was known by Euler in 1765, but rediscovered by Feuerbach in 1822.

WHY A STUDENT FAILS ???

Hello All
One of my student has send me this... Although I am not convinced... What do you say?

It's not the fault of the student if he fails, because the year has ONLY 365' days.

Typical academic year for a student.

1. Sundays-52,Sundays in a year, you know Sundays are for rest.
Days left 313.

2. Summer holidays-50 where weather is very hot and difficult to study.
Days left 263.

3. 8 hours daily sleep-means 130 days.
Days left 141.

4. 1 hour for daily playing-(good for health) means 15 days.
Days left 126.

5. 2 hours daily for food & other delicacies(chew properly & eat)-means 30days.
Days left 96.

6. 1 hour for talking (man is a social animal)-means 15 days !
Days left 81.

7. Exam days per year atleast 35 days.
Days left 46.

8. Quarterly, Half yearly and festival (holidays)-40 days.
Balance 6 days.

9. For sickness at least 3 days.
Remaining days 3.

10. Movies and functions at least 2 days.
1 day left.

11. That 1 day is your birthday. "How can you study at that day?"
Balance days 0

"How can a student PASS?"

Well... Can you tell me whats wrong in the arguments given above. Do Respond...

August 7 , 2009 is special

At 12hr 34 minutes and 56 seconds on the 7th of August this year, the time and date will be 12:34:56 07/08/09
Amazing!!!
1 2 3 4 5 6 7 8 9
This will never happen in your life again!!!

Small truth to make our Life 100% successful....

Let A = 1,B = 2, C = 3, D = 4, E = 5, F = 6, G = 7, H = 8, I = 9, J = 10, K = 11, L = 12, M = 13, N = 14, O = 15, P = 16, Q = 17, R = 18, S = 19, T = 20, U = 21, V = 22, W = 23, X = 24, Y = 25, Z = 26
______
Then,

H = 8
A = 1
R = 18
D = 4
W = 23
O = 15
R = 18
K = 11

Makes 98%
______

K = 11
N = 14
O = 15
W = 23
L = 12
E = 5
D = 4
G = 7
E = 5

Makes 96%
_____

L = 12
O = 15
V = 22
E = 5

Makes 54%
_____

L = 12
U = 21
C = 3
K = 11

Makes 47%

(None of them makes 100%)
_____

Then what makes 100%??

Is it Money? ..... NO!!!

Leadership? ..... NO!!!

Every problem has a solution, only if we perhaps change our "ATTITUDE".

It is OUR ATTITUDE towards Life and Work that makes OUR Life 100% Successful..

A = 1
T = 20
T = 20
I = 9
T = 20
U = 21
D = 4
E = 5
Makes 100%

Napoleon's Theorem

Take any triangle, and construct equilateral triangles on each side whose side lengths are the same as the length of each side of the original triangle.

Surprise: the centers of the equilateral triangles form an equilateral triangle!

This theorem is credited to Napoleon, who was fond of mathematics, though many doubt that he knew enough math to discover it!

Goldbach's Conjecture

Here's a famous unsolved problem: is every even number greater than 2 the sum of
2 primes?

The Goldbach conjecture, dating from 1742, says that the answer is yes.

Some simple examples:
4=2+2, 6=3+3, 8=3+5, 10=3+7, ..., 100=53+47, ...

What is known so far:
Schnirelmann(1930): There is some N such that every number from some point onwards can be written as the sum of at most N primes.
Vinogradov(1937): Every odd number from some point onwards can be written as the sum of 3 primes.
Chen(1966): Every sufficiently large even integer is the sum of a prime and an "almost prime" (a number with at most 2 prime factors).

Try it! Its really very interesting.

Well! Can you prove or disprove Goldbach’s conjecture.

Four Fours Problem

Here's a challenge that you may wish to try: can you express all the numbers from 1 to 100 using an arithmetic combination of only four 4's?

The operations and symbols that are allowed are: the four arithmetic operations (+,x,-,/), concatenation (44 is ok and uses up two 4's), decimal points (using 4.4 is ok), powers (using 4⁴ is ok), square roots, factorials (using 4! is ok), and overbars for indicating repeating digits (e.g., writing .4 with an overbar would be a way of expressing 4/9). Ordinary use of parentheses are allowed. No digits other than 4 are allowed.

This problem is sometimes called the four fours problem. It was popularized by Martin Gardner, among others.

For Example:

So... Take out your pen and paper... find all the possibilities from 1 to 100. Write your answers in the comment box.

Complex Roots Made Visible

Everyone learns that the roots of a polynomial have a graphical interpretation: they're the places where the function crosses the x-axis. But what happens when the equation has only imaginary roots? Do those have a graphical interpretation as well?

Here's an interpretation that works for quadratics. Take a quadratic, such as and graph it. In the given figure, it is shown in red. Because it lies entirely above the x-axis, we know it has no real roots.
Now, reflect the graph of the quadratic through its bottom-most point, and find the x-intercepts of this new graph, shown in green. Finally, treat these intercepts as if they were on opposite sides of a perfect circle, and rotate them both exactly 90 degrees. These new points are shown in blue.
If interpreted as points in the complex plane, the blue points are exactly the roots of the original equation! (In our example, they are 2+i and 2-i.)

Source: Mudd Math fun facts

Bike with Square Wheels

Ride a bike with round wheels and it rolls smoothly on a flat road (i.e., the axle remains level). Can you get a bike with square wheels to roll smoothly on a road of some other shape?

Surprisingly, yes. A road made up of inverted catenaries will do the trick! A catenary is a portion of a cosh curve (higher Mathematics). Figure below shows how such a bike would roll.

(Source: Mudd Math fun fact)