A GIFT THAT A KING CANNOT DELIVER

There was a chess competition in a king's regime. The king promised gift of winner's choice. After the game, the winner just asked some grains of rice as his price. He added one condition.
He wanted to put one grain of rice on the first square of chess board; 2 grains on the 2nd square; 4 grains on the 3rd; 8 grains on the 4th one. That is, doubling the number of grains each time till reaching 64th square. The king was surprised by the humble request and ordered to execute the prize. The servants brought sack full of rice and begum to arrange the grains on the board as instructed. Halfway through the work, the king and the servants realized that all rice grains in his kingdom would not be enough to fill all the 64 squares. The king submitted to the wisdom of the winner. Then the winner said, "I do not want the price. Just give rice food to the needy daily". And he went away.

This is a good example for the 'geometric series' which is of the form,

= a+ar+ar^2+ar^3+............+ar^n

our example,
=1+1*2+1*4+1*8+............
=1+1*2+1*2^2+1*2^3+.......+1*2^64
The sum of series = a[1-r^n/1-r]
hence, no.of grains= 1*[1-2^64/1-2]
= [1-2^64]/-1 = -1+2^64
= 2^64-1
= 18446744073709551615 grains
=460 billion tones of rice
Moral of the story: the mathematics can kill our ego.

Is 0.999......equal to 1:

0.999....... = 0.9+0.09+0.009+0.0009+..............
= 9/10+9/100+9/1000+.....................
=9/10[1+1/10+1/10^2+1/10^3+.........]-------A
so the series in the bracket is geometric series with a=1,r=1/10 and n goes to infinity
we know,The sum of series = a[1-r^n/1-r]
= 1*[1-(1/10)^infinity/1-1/10]
1/10 to the power infinity logically becomes zero.
than, sum = 1/[1-1/10]
= 1/[9/10] = 10/9
multiply the sum 10/9 with 9/10 which was outside the bracket in equation A
FINALLY,
0.999.......= [9/10]*[10/9] = 1
0.999........definitely is equal to 1

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LISSAJOUS FIGURES

Definition: "When a particle is subjected to two sine wave motion or two oscillatory motion at right angles, the particle describes lissajous figures". We know sine wave motion and circular motion is basically same. Hence we draw two circles A and B perpendicular to each other. The circle B rotates twice faster than circle A. That is, frequency of circle B is two times than that of A. A particle at the intersection of two circles is subjected to two sine wave motion A and B at 90 degree simultaneously. The particle will describe figures depending on the frequency and phase of A and B . In our case, the ratio of frequency is 1:2 and the two waves are in phase. To draw lissajous figures : A moving point in both the circles are chosen. Here we should remember; during the time taken by the circle A to complete one rotation, circle B completes two. Hence the points are marked on the circles according to their speed. Then straight lines

THE PARABOLA

A jet of water shooting from a hose pipe will follow a parabolic path. What is the so special about parabola. Y= x^2 Draw a graph for the above equation. It will result in a parabola. This parabola is also called unit parabola. Any equation involving square will yield a parabola. Example: Y = 2x^2 +3x+3 (also called quadratic equation) X= 2 and -2, both satisfies the equation 4 = X^2. Parabolic equations always have two solutions. Any motion taking place freely under gravity follows parabolic path. Examples: An object dropped from a moving train, A bomb dropped from flying plane, A ball kicked upwards. If a beam of light rays fall on the parabolic shaped mirror, they will be reflected and brought to focus on a point. This fact is made use of in Dish Antenna, Telescope mirrors, etc. Inverted parabola shape is used in the construction of buildings and bridges. Because the shape is able to bear more weight. A plane

CASINO'S GAME

Let us find out how the casino survives with mathematics. Say, your friend invite you for a game of dice. You must bet (wager) 2 dollars. If you roll 'six' you will get back 8 dollars. The game will go on for 30 rounds. All sounds good. The probability of rolling 'six' is 1/6. Since the game will be played for 30 times, the 'expected win' is 30*1/6 = 5. That is, you are expected to win 5 rounds out of 30. Hence your gain will be 5 * 8 =40 dollars. ok. This also implies that you will loose 25 rounds. Hence your loss will be 25*2 =50 dollars. Your net gain will be gain-less = 40-50 = -10 dollars. For 30 rounds, the loss is -10 dollars, Hence, for one round =-10/30 = -1/3 dollars. There will be a loss of -1/3 or 0.33 dollars per round. It is not a fair game. Let us make a simple formula to calculate 'Pay out per round\. The probability for a win = p The pay-out in case of win = V No. of rounds = n The expect

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