I'm teaching geometric series to my January maths students this week. I already subjected them to Zeno's paradox of Achilles and Tortoise in the lecture earlier in the week (here's a one minute video narrated by David Mitchell which will tell you all you need to know about it). Then we did some "proper" exam style questions on the subject in the seminars (first term, common ratio, sum to infinity, etc).

But I can't resist the story telling....and today my seminar group got the apocryphal story of the invention of chess. In my rambling, inaccurate version which is set "long ago in a country far away" the invention of chess pleases the King of the country so much that he decides to reward the inventor with whatever he desires. The inventor being a wily mathematician with not much imagination tells the King he wants some rice. But how much rice? Well, put one grain on the first square of the chess board, he says. Two grains on the second square. Four on the third square. Eight on the fourth square. And so on. The King, no doubt sensing a bargain, readily agrees to this.

At this point I ask my students - how many grains of rice will be on the final square of the board? And how many rice grains in total on the chess board?

That old King might have been fooled but my students weren't (well, not all of 'em), The number of rice grains doubles each time - the sum is a geometric series. The number of rice grains on square \(n\) is given by \(2^{n-1}\). That means the total number of grains on the final square of the board is \(2^{63}\). A quick calculation using the sum formula for \(n\) terms of a geometric series showed that the inventor was owed a staggering \[2^{64} -1\] grains of rice.

That's a huge number. Bigger....much, much bigger than millions, billions or trillions. I asked my students to find a way to convey the sizes of these numbers to a non-mathematician (not me, cheeky reader). The lesson went off on a bit of a tangent as they searched for ways to bring meaning to that amount of rice. We decided to calculate the total mass of the rice on the board. We asked Wolfram Alpha how much a single rice grain weighed and it came back with 65mg. Sounds about right. My students did some quick conversions and it turned out that the inventor had laid claim to around \(6\times 10^{14}\) kg of rice.

Anyway, that's still a big number and hard to visualise. The students changed it to tonnes and it was about \(6\times 10^{11}\) tonnes (or 600 billion tonnes).

Someone suggested expressing it in terms of bigger objects. An elephant is big. But we decided that an elephant weighed about 1 tonne....so that didn't help. Think bigger.

A bit more head scratching and they settled on the Airbus A380 as a unit of mass. Back to Google and Wikipedia and it turns out an Airbus A380 weighs in at 650 tonnes. That means the mountain of rice on the chess board has the same weight as 922 million Airbuses. It's still not easy to picture.

A bit more head scratching and they settled on the Airbus A380 as a unit of mass. Back to Google and Wikipedia and it turns out an Airbus A380 weighs in at 650 tonnes. That means the mountain of rice on the chess board has the same weight as 922 million Airbuses. It's still not easy to picture.

We changed tack a little. Suppose we could take all that rice and make a mountain out of it. We'd squash it together and compact it to the density of rock. How big would Rice Mountain be?

We chose a mountain shaped like a right-circular cone. We just had to figure out how big the cone should be. We needed to change our mass into volume at the density of rock. Back onto Google to get the density of rock....we settled on a figure of around 3000 kg per cubic meter. Every 3000 kg of rice would get us 1 cubic meter of mountain. They calculated the volume that the compacted rice would occupy: it turned out to be \(2\times 10^{11} m^3\).

The volume of a cone with radius \(r\) and height \(h\) is \(\tfrac{2\pi r^2 h}{3}\).

To make things easier we decided the height \(h\) of the mountain would be equal to the radius \(r\). That meant the cone/mountain volume was \(\tfrac{2\pi h^3}{3}\).

Plugging in the numbers and rearranging gave us a mountain height of 4,570 metres...give or take a bit.

To make things easier we decided the height \(h\) of the mountain would be equal to the radius \(r\). That meant the cone/mountain volume was \(\tfrac{2\pi h^3}{3}\).

Plugging in the numbers and rearranging gave us a mountain height of 4,570 metres...give or take a bit.

Now that's a number we can visualise. Rice Mountain is about half the height of Mount Everest (but soooo much harder to climb its steep, perfectly smooth face).

These calculations took about ten minutes to do in class. I think some of the students were a little bemused by it ("Will we get questions like this in the exam?") but most were going with it and were happy that they could tame enormous numbers by making them into mere mountains.

Who knows what the King thought of the chess inventor when he realised he'd been duped. Perhaps he clapped him on the back, wished him well and sent him on his way with his head still on his shoulders. Hopefully not.