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     There is a famous story about a king, a chessboard, and a lot of rice that you may have heard about. A king wanted to give a servant a reward for an act of heroism. He asked the servant what he wanted. He probably laughed when the servant gave his request. He wanted the king to put one grain of rice on the first square of a chessboard on the first day, and two on the second square on the second day, four on the third, eight on the fourth, and so on. Each day doubling the number of grains of rice from the day before.

     The king agreed, thinking that this poor servant had wasted his wish. Before the end of the sixty-four days he definitely felt differently. The amount of rice required to pay this debt would have been more than the total rice production of earth.

On the first square of the chessboard the king would place one grain of rice. On the second he would place two. Then four, eight, sixteen, etc. We could keep doubling this way until we got to the very last square.

But we don't want to know how many grains of rice are on the last square, we want to know how many grains of rice are on all of the squares combined. Of course, we could just add them all together, but would you believe me if I said that to find that answer all we would have to do is double that final number one more time and subtract one? Let's see if I can prove it to you.

Now that you understand how it works, let's talk about how we write it down. Trust me, it makes things much easier as problems get more complicated.

Is how we annotate x to the power of y. That means we multiple y x's together. So:

(Quick note: any non-zero number to the power of zero is one.)

Now to make this easier, I am going to call the first square square zero (we often start with zero in math.) So we can write the sum of our series as:

Both sides of that equation basically mean:

Add up all of the 2 to the power of n from zero to 63. 

Now that you know the trick from above, we know that the answer is:

Now is the time to break out your calculators! So the servant would have 18,446,744,073,709,551,615 grains of rice on his chessboard. 

Scientific notation makes this number easier to think about.

That would be written: 1.8x10^19 or 1.8 times 1 with 19 zeros. 

It's a big number, but how big?

Not as many as the estimated stars in the universe (~2x10^23)

But more than the estimated grains of sands on earth (~7.5x10^18)

And how much rice is it?

One pound is equal to 453.6 grams. There are about 48 grains of rice in one gram. 

So one pound = 453.6 x 48 = 21,773 grains.

So the chessboard has 847,230,242,672,555 pounds of rice or 423,615,121,336 tons.

Now we are finally getting to numbers that are easier to say. That chessboard has more than 400 trillion tons of rice on it.

The total rice production of the world is about 780 million tons a year. So at current levels, it would take the whole world 543 years to pay off the king's debt!