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Chessboard and pyramid


Have you heard of the legend of the chessboard? The story goes like this. There was a wise man who invented the chess game and introduced it to a king. The king loved the game so much that he told the wise man that he could choose anything for a reward. The wise man then pointed to the chessboard and asked for 1 grain of wheat for the first chess square, 2 grains of wheat for the second square, 4 grains of wheat for the third square, 8 grains of wheat for the fourth square, and repeat this doubling pattern. This sounds like a very little reward but at the end the king didn't have enough wheat to reward the wise man.


Today, we will calculate to see how much wheat the wise man actually requested. Is it a lot? Is it little? I'll give you a clue, it's HUGE!

To see how big this reward is, we will calculate the number of pyramids that we could build up using all this wheat.




Step 1. Find the total number of grains

A chessboard has $8 \times 8 = 64$ squares and the wise man asked for his reward the following:
  • Square 1: number of grains $=1$
  • Square 2: number of grains $=2$
  • Square 3: number of grains $= 4=2^2$
  • Square 4: number of grains $= 8=2^3$
  • Square 5: number of grains $=16=2^4$
  • Square $n$: number of grains $=2^{n-1}$
  • Square 63: number of grains $=2^{62}$
  • Square 64: number of grains $=2^{63}$

So the total number of grains is $$S = 1 + 2 + 4 + 8 + 2^4 + \dots + 2^{62} + 2^{63}$$

We will simplify the formula $S$ as follows. Multiply both sides of the formula $S$ by 2, we have $$2S = 2 + 4 + 8 + 16 + 2^5 + \dots + 2^{63} + 2^{64}$$
Subtract it to the original formula $$S = 1 + 2 + 4 + 8 + 2^4 + \dots + 2^{63}$$ we obtain $$S = 2^{64} - 1$$


Step 2. Estimate the volume of the wheat

We have calculated that the total number of grains is $S = 2^{64} - 1$. For simplicity, we will give the wise man one more grain of wheat to make $S = 2^{64}$. We will assume that a grain of wheat has a volume of 2 cubic millimetres so that the total volume of all the wheat is $V = 2^{65}$ cubic millimetres.

But how can we estimate the number $2^{65}$?

In computer science, we have
  • 1 kilobyte = $2^{10}$ bytes $\approx$ 1000 bytes,
  • 1 megabyte = $2^{10}$ kilobytes $\approx$ 1000 kilobytes,
  • 1 gigabyte = $2^{10}$ megabytes $\approx$ 1000 megabytes,
  • 1 terabyte = $2^{10}$ gigabytes $\approx$ 1000 gigabytes.

Thus, we have made the following approximation: $$2^{10} = 1024 >\approx 1000$$

Now we will use the same approximation $2^{10} \approx 1000$ to estimate the volume $V = 2^{65}$ (cubic millimetres): $$V = 2^{65} = 2^{5} \times 2^{60} = 32 \times 2^{60} >\approx 32 \times 1000^{6}$$
Therefore, $V \approx 32 \times 1000^{6}$ cubic millimetres $=32 \times 1000^{3}$ cubic metres.


The volume of all the wheat that the wise man asked for his reward is
$V\approx$ 32 billion cubic metres!




Step 3. Estimate the volume of a pyramid


The pyramids in the above picture are the Giza pyramids. They are located in the outskirts of Cairo - capital of Egypt. The pyramids are the burial tombs of Egypt pharaohs. The pyramid on the far side is the largest pyramid, 147 m tall, tomb of a pharaoh named Khufu. It was constructed first around 2550BC. The middle pyramid, 144 m tall, 2520 BC, was the tomb of a pharaoh named Khafre. The one in the front is the smallest, 65 m tall, tomb of a pharaoh named Menkaure, constructed around 2490BC.

We have a formula to calculate the volume of a pyramid. If a pyramid has a height of $h$ and its base is a square of side length $a$ then its volume is $$V = \frac{1}{3} a^2 h$$

Apply this formula, we have:
height hbase length avolume V
Pyramid of Khufu147 m230 m2,592,100 cubic metres
Pyramid of Khafre144 m215 m2,218,800 cubic metres
Pyramid of Menkaure65 m105 m238,875 cubic metres

The largest pyramid among these three, the pyramid of Khufu, has a volume of approximately 2.6 million cubic metres.



Step 4. Find the number of pyramids

Now we are ready to answer the question that we put forth in the beginning. We have estimated the volume of the wheat that the wise man requested: 32 billion cubic metres. We have calculated the volume of the largest Giza pyramid: 2.6 million cubic metres. If we could use all this wheat to form pyramids then the number of wheat pyramids would be $$\frac{32 \times 1000^3}{2.6 \times 1000^2} > \approx 12000$$
So the wise man had requested a gift of more than 12 thousand Giza pyramids of wheat. No wonder why the king did not have enough wheat to reward the wise man!!!


Let us stop here for now. The above pictures of Giza pyramids were taken from Google Maps. The picture of a chessboard at the top of our post is a screenshot from a movies by Cristóbal Vila - "Inspirations" - follow the link below to watch it on YouTube:




Homework.


1. Go to google.com and find out how large the moon surface is. Is it possible to cover the whole surface of the moon by the wise man's wheat?


2. Simplify the sum $$1 + \frac{1}{2} + \frac{1}{4} + + \frac{1}{8} + \dots + + \frac{1}{2^{64}}$$

3. Simplify the sum $$1 + 3 + 9 + 27 + \dots + 3^{64}$$



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