 An old legend from India, about the origins of chess tells the story that more than 1,000 years ago, an inventor created chess for his king. Being very pleased, the king offered a reward. Let us look at the inventor's request. He suggested giving me grains of wheat upon each square of the chess board such that one grain is placed on the first square, two on the second, four on the third and so on, doubling the number of grains on each subsequent square. Let us look at this more carefully. So he wanted one grain to be placed here, two grains here, four here, and so on. So let us write age, 16, 32, and so on, 64, 28, until at the very end you would have noticed that this is just 2 to the power of 0, 2 to the power of 2, 2 to the power of 3 and so on. And this one would be, since we are starting with 2 to the power of 0, this one would be 2 to the power of 63. This would be equal to 1 plus 2 plus 2 square plus 2 cubed and so on plus 2 to the power of 63 which is equal to 2 to the power of 64 minus 1. When the king realized that the inventor had tricked him and that there was not enough wheat in the entire kingdom to give to the inventor, the king ordered the inventor killed. Now let us take a closer look at this computation that we just made. What is it? That 1 plus 2 plus 2 square plus 2 cubed and so on plus 2 to the power of 63 is equal to 2 to the power of 64 minus 1. It's a very simple trick that we are going to use. Basically we know that if we have an amount, we can double that amount and then take away the original amount. We are going to use that very simple trick. So we have this amount. Let's just write it like this, 2 to the power of 0 plus 2 plus 2 squared and so on. What we are going to do is, we are first going to multiply this by 2, that amount, and then let's write 62 here and then power of 63 and then we have to subtract the original amount which is 2 to the power of 0 minus minus and so on minus, because we are subtracting all those quantities, that's correct, but we are here, we are inside this parenthesis. So this is 2 to the 63rd power. So when you do this, you end up with this, when you carry out that multiplication, you are going to take away this. Notice that we are subtracting those terms so this is going to cancel with this one. These two cancel out, these two cancel out and so on and these two cancel out and we are left with only these two terms, 2 to the power of 64 minus 1. Thank you.