CHOCOLATE PUZZLE

A shop sells chocolates @ Re.1 each. You can exchange 3 wrappers for 1 chocolate.


If you have Rs.15/- how many chocolates can you get totally????
 
 
Find the solution..
 
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SOLUTION:
 
22 CHOCOLATES

 for  15 rupees you get 15 chocolates.
on returning 15 wrappers you get 5  chocolates.
on returning 3 wrappers  (keeping two wrappers in hand) you get 1 chocolate.
now this one  chocolate. wrapper and the twp wrapers you had will add on three.
atlast on returning 3 wrappers you get 1 chocolate.


i.e.,   15+5+1+1 = 22

MATHS AND NATURE

"The laws of nature are but the mathematical thoughts of God"

                                                                                      - Euclid

Mathematics is everywhere in this universe. We seldom note it. We enjoy nature and are not interested in going deep about what mathematical idea is in it. Here are a very few properties of mathematics that are depicted  in nature.

SYMMETRY

Symmetry is everywhere you look in nature .

Symmetry is when a figure has two sides that are mirror images of one another. It would then be possible to draw a line through a picture of the object and along either side the image would look exactly the same. This line would be called a line of symmetry.

There are two kinds of symmetry.

One is bilateral symmetry in which an object has two sides that are mirror images of each other.

The human body would be an excellent example of a living being that has bilateral symmetry.




Few more pictures in nature showing bilateral symmetry.

The other kind of symmetry is radial symmetry. This is where there is a center point and numerous lines of symmetry could be drawn.

The most obvious geometric example would be a circle.

Few more pictures in nature showing radial symmetry.

SHAPES

Geometry is the branch of mathematics  that describes shapes.

Sphere:

A sphere  is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball.

The shape of the Earth is very close to that of an oblate spheroid, a sphere flattened along the axis from pole to pole such that there is a bulge around the equator.

Hexagons:

Hexagons are six-sided polygons, closed, 2-dimensional, many-sided figures with straight edges.

 
For a beehive, close packing is important to maximise the use of space. Hexagons fit most closely together without any gaps; so hexagonal wax cells are what bees create to store their eggs and larvae.

Cones:

A cone is a three-dimensional geometric shape that tapers smoothly from a flat, usually circular base to a point called the apex or vertex.

Volcanoes form cones, the steepness and height of which depends on the runniness (viscosity) of the lava. Fast, runny lava forms flatter cones; thick, viscous lava forms steep-sided cones.

Few more cones in nature:

Parallel lines:

In mathematics, parallel lines stretch to infinity, neither converging nor diverging.

These parallel dunes in the Australian desert aren't perfect - the physical world rarely is.

Fibonacci spiral:

If you construct a series of squares with lengths equal to the Fibonacci numbers (1,1,2,3,5, etc) and trace a line through the diagonals of each square, it forms a Fibonacci spiral.

Many examples of the Fibonacci spiral can be seen in nature, including in the chambers of a nautilus shell.

 
 
 
 
 
 
 
 
 
 
 
 

VAMPIRE NUMBER

The vampire numbers were introduced by Clifford A. Pickover in 1994.

A Vampire number z  is a number which can be written as a product of two numbers x and y  containing the same digits the same number of times as the vampire number. Here x and y are called FANGS.

FOR EXAMPLE:

1827 can be written as a product of two numbers 21 and 87.
i.e., 1827 = 21 * 87
Here the digits 1,8, 2,7 are repeated  the same number of times but in different order. 21 and 87 are called fangs.

A true Vampire number isa number which can be written with 2 fangs having the same number of digits not both ending in zero.

FOR EXAMPLE :

1260 is a vampire number, with 21 and 60 as fangs, since 21 × 60 = 1260. However, 126000 – 210 × 600 – is not, as both 210 and 600 have trailing zeroes.

All vampire numbers must clearly have an no. of digits.


Some vampire numbers:
 117067 = 167*701
124483 = 281 * 443
536539 = 563 · 953
23287176 = 2673 * 8712.




Vampire number having two distinct pairs of fangs :


125460 = 204 *615 = 246 * 510

Vampire number having three distinct pairs of fangs :

13078260 = 1620*8073 = 1863*7020 = 2070*6318

SIMPLE MATH PUZZLES

PUZZLE 8

What mathematical symbol can be put between 5 and 9, to get a number bigger than 5 and smaller than 9?

PUZZLE 9

The following multiplication example uses every digit from 0 to 9 once (not counting the intermediate steps). Fill in the missing numbers.

7 _ _  *   4 _  =  _ _ _ _ _

PUZZLE 10

Here's a simple multiplication problem in which each letter represents a different digit. Can you solve it?

IF * AT = FIAT

PUZZLE 11:

Write down the next number in this series:

18,46,94,63,52, ?

PUZZLE12:

Find the next letter in the series:

O,T,T,F,F,S,S,E,?

PUZZLE 13:

How many two-digit positive whole numbers are there?

PUZZLE 14:

Find four consecutive prime numbers that add up to 220.

SOLUTIONS

 
PUZZLE 8 : A Decimal Point. 5.9
 
PUZZLE 9 : 715 * 46 = 32890
 
PUZZLE 10 : IF = 41. AT = 35. FIAT = 1435.


PUZZLE 11 : 61. Each is a perfect square read from back to front


PUZZLE 12 : N –  The series is the first letter of the numbers 1, 2,3,4,5,6,7,8, ? .so the next letter is N the first letter of Nine.  1[One],2[Two], 3 [Three], … 8[Eight], 9 [Nine].


PUZZLE 13: 90


PUZZLE 14:  47,53,59,61.
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FASCINATING FACTS OF MATHEMATICS: PERMUTATIONS AND COMBINATIONS

FASCINATING FACTS OF MATHEMATICS: PERMUTATIONS AND COMBINATIONS

PERMUTATIONS AND COMBINATIONS

In common, students who come across the topic “Permutation and Combination” feel it to be confusing.

The basic idea of the topic is selection and arrangement of things. They are widely used in solving problems of probability, genetic engineering and life sciences.

Combination to unlock a safe.. Consider a 3 digit number 653 which you use as a combination to the safe. Here the order of the number is important as 563 will not work to unlock the safe. This is permutation where the order of elements is very important.

Combination of three colors.. Consider 4 colors red, blue, yellow and green. If I ask you to select 3 colors and you choosing them as Red, green and blue is same when you choose it as blue, red and green. This is combination where the order does not matter.

To make it simple, mathematically it can be stated as:

“Permutation is an ordered Combination”.

An easy way to memorize:

Permutation  - Position important

Combination - Chosen important

Permutation:

Permutation is arrangement of values in all possible ways.The word arrangement is used to emphasize that the order of the things is important.

Consider the three letter word “PEN”. In how many ways can we rearrange it.

       PEN   PNE   EPN   ENP   NEP   NPE

A three letter word can be arranged in 6 ways. You can easily find it out. How about a 10 letter word. Quite a long process isn’t it. Hence we use FACTORIAL to find the number of possible ways to arrange the elements.
For a 3 letter word,

         3! = 3*2*1 = 6 ways.

For a 10 letter word,

10! = 10*9*8*7*6*5*4*3*2*1= 3628800

Permutation with Repetition:

Number of permutations of n-things, taken ‘r’ at a time when each thing can be repeated r-times is given by = n ^ r.

A child has 3 pocket and 4 coins. In how many ways can he put the coins in his pocket.
First coin can be put in 3 ways, similarly second, third and forth coins also can be put in 3 ways.

So total number of ways = 3 x 3 x 3 x 3
                                        = 3 ^ 4 = 81

Permutation without Repetition:

If we have to select 3 numbers from a set of 9 numbers say, from 1 to 9 , then what are the possibilities?

For the first number we have 9 choices{1,2,3,4,5,6,7,8,9}. Let us choose 4.

For the second number we have 8 choices{1,2,3,5,6,7,8,9}. Let us choose 8.

For the third number we have 7 choices{1,2,3,5,6,7,9}. Let us choose 3.

We had 9 choices at first, then 8 and then 7.Therefore the total no. of options would be 9*8*7 = 504.

So, if you wanted to select all of the 9 numbers, the permutation would be:

9! =9*8*7*6*5*4*3*2*1 = 362880.

But here you wanted to select just 3, then you have to stop the multiplying after 7. How do you do that? There is a simple trick ... you divide by 6! ...

9! ÷6! = (9*8*7*6*5*4*3*2*1) / (6*5*4*3*2*1)
          = 9*8*7 = 504.

Mathematically, the number of permutations of ‘r ‘objects chosen from a set of ‘n’ objects is expressed as:

  nPr = n! / (n-r)!

Combination:

Combination means selection of things. The word selection is used, when the order of things has no importance.

            If 9 players are selected to form a team from 20 players, the order in which the 9 players are selected doesn’t matter as they are all in the team. This is combination. There will be a change in combination iff a player in the team is changed.

In permutation the order is important as 234 is different from 324. but in combination we are concerned only with the numbers 2, 3 and 4 that have been selected. The combination of 234 is same as 324.

The combinations of abcd taken 3 at a time :

              abc abd acd bcd.

Each of these four combinations will give rise to 3! Permutations:

    abc     abd     acd     bcd

    acb      adb     adc     bdc

    bac      bad     cad     cbd

    bca      bda     cda     cdb

    cab      dab     dac     dbc

    cba     dba     dca     dcb

Each column is the permutation of that combination. But they are all one combination as order has no importance.Since the order does not matter in combinations, there are clearly fewer combinations than permutations. Hence combinations can be stated as :

"Combinations are subsets of permutation".

Thus if we want to figure out how many combinations we have of n objects then r at a time, we just create all the permutation and then divide by r! variant.

Mathematically, the number of combinations of n things taken r at a time is:

nCr = nPr / r!

                 = n! / [r! (n- r)!]

nCr is known as n choose r. It is also known as Binomial co-efficient.

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STATISTICS IN DAILY LIFE

• Statistics a mathematical science form a basic tool in business and manufacturing as well.

• MEAN:

If in a tour, the total money spent by 10 friends is Rs.50000, then the average money spent by each person is Rs.5000.Here 5000(50000 /10 ) is the mean.

• MEDIAN:

If you have 15 things lined up next to each other by their cost, the median cost will be the cost of the thing in the very middle. Here the cost of middle thing is the median.

• MODE:

A shopkeeper, selling shirts, keeps more stock of that size of shirt which has more sale. Here the size of that shirt is the mode among other.

MATHS IN COOKING

Cooking is an art. While preparing a delicious meal, measuring the ingredients and calculating the time it will take to cook involve maths. If a person adds too much of salt or sugar it will spoil the food. One must be aware in what proportion it should be added. Converting measurements when doubling and tripling a recipe needs basic math skills of multiplication. Knowing how to measure can make the difference between a great meal and a horrible. People who enjoy cooking understand that they need to have sharp math skills in order to prepare good food.

MATHS IN DAILY LIFE

The language of mathematics is numbers and it is the only language shared by all people regardless of culture , religion or gender.

Roger Bacon, an English philosopher and scientist stated :

"Neglect of mathematics works injury to all knowledge, since he who is ignorant of it cannot know the other sciences or the things of the world."

Adding up the cost of a basket full of groceries involves the same math process regardless of whether the total is expressed in dollars, rubles, or yen.With this universal language, all of us, no matter what our unit of exchange, are likely to arrive at math results the same way.Math can help us to shop wisely, buy the right insurance, remodel a home within a budget, understand population growth, or even bet on the horse with the best chance of winning the race.

ZERO

ZERO

... a fine and wonderful refuge of the divine spirit - almost an amphibian between being and non being.     --- Gottfried Leibniz

• The first thing to say about zero is that the use of zero is  extremely important .

It is used  as an empty place indicator in our place-value number system. Hence in a number like 1502 , the zero is used so that the positions of the 1 and 5 are correct. Clearly 152 means something quite different

• The name "zero" derives ultimately from the Arabic sifr which also gives us the word "cipher".

•  The first use of the symbol which we recognise today as the notation for zero "0"  is omicron, the first letter of the Greek word for nothing namely "ouden".

• THE FIRST RECORD OF INDIAN USE OF ZERO:

We have an inscription on an stone tablet which contains a date which translates to 876. T he inscription concerns the town of Gwalior , 400 km south of Delhi, where they planted a garden 187 by 270 hastas which would produce enough flowers to allow 50 garlands per day to be given to the local temple. 

Both of the numbers 270 and 50 are denoted almost as they appear today although 0 is smaller and slighty raised.

•    INDIAN MATHEMATICIANS

Brahmagupta attempted to give the rules for arithmetic involving zero and negative numbers in the seventh century.


 He gave the following rules for addition which involve zero:-


The sum of zero and a negative number is negative, the sum of a positive number and zero is positive, the sum of zero and zero is zero.

He gave the following rules for Subtraction  which involve zero:-
 

A negative number subtracted from zero is positive, a positive number subtracted from zero is negative, zero subtracted from a negative number is negative, zero subtracted from a positive number is positive, zero subtracted from zero is zero. 

Brahmagupta was not able to explain clearly about division.

Bhaskara tried to solve the problem by writing n/0 = ∞.

 If this were true then 0 times ∞ must be equal to every number n, so all numbers are equal.

Bhaskara did correctly state other properties of zero such as  

   0 ^ 2 = 0  and √ 0 = 0.

The Indian mathematicians could not bring themselves to the point of admitting that one could not divide by zero.

• The number zero is neither positive nor negative, neither a prime number nor a composite number, nor it is a unit. It is a even number.