HOW TO BEAT THE COMPUTER?

222,222,222,222,222,222^2 -222,222,222,222,222,222,221^2 =?

A calculator or a computer cannot solve this problem. But we can.

We know 1^2-0^2 =1

2^2-1^2=3

3^2-2^2=5

4^2-3^2=7

There seems to be a pattern here. To find the difference between two adjacent squares, you just have to add the unsquared numbers together. We are not sure that this pattern goes on forever.

Let us draw some pictures.

each square is made from the previous one. To go from 2^2 to 3^2, add 2 dots to the bottom and 3 dots to the side. The difference between 3^2 and 2^2 is 3+2=5. It is clear from the pictures.

Similarly, to go from 3^2 to 4^2, add 3 dots to one side and 4 dots to another side. The difference here is 3+4=7. The picture shows the same. Now we understand that the pattern will go on forever.

This is proof by induction. In the same lines, we can get the answer to our problem. Just add the two numbers without squaring.

222 222 222 222 222 222 222 +222 222 222 222 222 222 221= 444 444 444 444 444 444 443

got it.

The essence of "proof by induction":

step1. Show it is true for the first one.

step2: show that if anyone is true then the next one is true.

Then all are true.

In other words: a Domino effect

Step1: The first domino falls.

step2: when any domino falls, the next domino falls

So all dominos will fall.

falls one by one

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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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