HOW much rice?

There’s a legend, so well-known that it’s almost a cliche, about the wise man who invented chess. When asked by the great king what reward he wanted, he replied that he’d be satisfied by a chessboard full of rice: one grain on the first square, two on the second, four on the third, doubling each time.

The king, of course, laughed at his modest demands, and told his people to make it so.

His people nervously told the king that actually, that was quite a lot of rice, and if he knew about his Core 2 geometric sequences, he wouldn’t have been so badly duped. After all, $S_n=\frac{a(1-r^n)}{1-r}$. Here $a=1$, $r=2$ and $n=64$, so that works out to $2^{64}-1$, which is 18,446,744,073,709,551,615 grains of rice altogether.

“Do we have that much rice?” asked the king.

“Well, sire, that’s $1.8\times {10}^{19}$ grains, and there are about $3.6\times {10}^6$ grains in a tonne.”

“So it’s, what, $0.5\times {10}^{13}$… five trillion tonnes?”

“Very good, sire.”

“Do we have that much rice?”

“I’m afraid not, sire - even looking far into the future, say in the early 21st century, that’ll be roughly the entire worldwide crop for a decade.”

“Oh.”

But how much area would that take up?

“One square centimetre of rice,” said the king’s people, “is about ten grains.”

“So 18 quintillion grains needs $1.8\times {10}^{18}{\text{cm}}^2$?”

“Yes, sire, although we should convert that into more sensible units.”

“Fine. A metre squared is 100… no! 10,000 centimetres squared, which takes us down to $1.8\times {10}^{14}{\text{m}}^2$.”

“Still a little… unwieldy, sire.”

“Fine. Let’s take it down another million by talking about kilometres squared, so it’s $1.8\times {10}^8{\text{km}}^2.$ Is that a big number?”

“It’s quite big, sire.”

“How big is the world?”

“The world, sire? I don’t have that information to hand - but we can work it out. The world’s circumference is about $4\times {10}^4$ kilometres, so its radius is that divided by $2\pi$.”

“$6.3\times {10}^3\text{km}$?” guessed the king, who’d had some Ninja training. ¹

“Yes, sire. So the surface area is…”

“$4\pi r^2$,” interrupted the king. “$r^2$ is about $4\times {10}^7$, so it’s roughly $5\times {10}^8{\text{km}}^2$.”

“One day, sire, someone will invent a machine that will answer such questions in an instant.”

“But of course, only a third of the Earth’s surface is land.”

“Correct, sire.”

“Which is $1.7\times {10}^8\text{km}$. So the wise man wants enough rice to cover pretty much every landmass on the planet to a depth of one grain. Fine. I think this can be easily solved.”

And the king had the wise man’s head chopped off.

The moral of the story

Nobody likes a smartarse.

* Thanks to Aidan for working this out with me.

Footnotes:

1. $4÷2\pi \simeq 4\times \frac{7}{44}=\frac{7}{11}=0.63$