1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 = 255 = 2⁸-1

A very long time ago, in an age from which folk have come, there was (allegedly) an Emperor of (course, of) China who found himself in the position of having to reward a peasant for saving the life of his daughter. This was instead (again, of course) of giving the hand of said daughter in marriage, presumably, since that is the way such stories are told. Feminists should retell the story with a currency of daughters, dowries and, applying some telling economic theory, reconstructing an equalised economy, preferably without slavery.

Unfortunately, the emperor hadn’t studied the maths that we have....

His generous reward started with one grain of rice on the first square of a chess board. Imagine he has gardens laid out as the chessboard, eight squares in each of two perpendicular directions if it helps you to visualise this scene. So the first grain is easily lost. On the second square go two grains of rice; on the third go four. Carry on doubling, and at the end of filling the first row there are

in total 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 = 255 = 2⁸-1 grains.

The grand vizier (oh come on, there has to be one at least) is becoming worried - he has begun to see what is coming. Don’t cast the vizier as a bad person, but one who ‘gets’ numbers….

Questions:

1. How many grains of rice for the last, the 64th, square?

2. How much rice is required for all the squares in total? Write this number down as a formula, and in standard form to 6 sig fig. Be warned, it is a BIG number.

3. Suppose, as we discussed in the first class to do this, a grain of rice is 1mm x 1mm x 7mm. How many grains make a litre? (That’s 10cm x 10cm x 10cm)

4. How many grains make a cubic metre? Again, it is a large number. Let’s bag up the rice, ALL of it, so the young man might take it away; it is China, we have lots of people for a labour market….. (this was well into the second lesson on this, possibly the second day, after they’d tried ideas out as homework)

5. If we treat our pile of rice in bags as a cuboid as high as the Main Building of school, that will be about 10metres high. So how many square kilometres is this? (I have the right unit, amazingly). If the pile had a square base, how long is a side? (Try kilometres; it is a really big pile).

6. Suppose we try piling the bags as a mountain. Treat this pile as a pyramid; The formula is one third of the base area times the height. Try setting the height the same as the side and see how big this dimension would be. Suppose we set the height to be as much as one kilometre (that’s the same as the highest peaks in Wales); now how long is one side of the mountain if we keep the base square?

7. Can you work out the diameter of the pile if we made it as a circular cone? (You might say “No, I can’t”).

8. North Dartmoor is about 180 sq km (from the Tavistock - Princetown road to Okehampton). If we (could ) spread our pile of rice sacks over this whole area, how deep would it be? That’s before it gets wet, when rice expands.

9. Apparently, when rice gets wet (hey, it IS Dartmoor we’re talking about) it swells to three to five times its size. This still causes ships to sink if they let their rice cargo get soaked — it breaks the hull and so sinks the ship. If the pile stays on the moor and grows only in height, how big is the pile now?

10. Extensions: these are scarily big numbers and the first class to explore this spent the whole week's lessons coming up with new things to try out. Do you think the Emperor had access to that much rice? Do we grow that much now? How many square kilometres of rice field (paddy fields) would we (the human race) need to crop that much rice? How do we work that out? Is Africa big enough? For extension, discuss with others, set up research groups, share out that research, involve older people — go get answers by finding out. But not by simply going to see if someone has worked this out for you. This is supposed to be interesting, and if you are not fired up by this, simply do not do it. Those who find this interesting have found a use for geography, for economics, for biology - and for maths. Well, arithmetic and a little modelling, at least.

DJS 20160909

based on something written 20050428

1. 2^{64} = 1.84467 x 10 ^{19}

2. 2⁶⁵ - 1 = 3.689348815 x 10^{19} = 3.68935 x 10^{19} to 6 s.f.

3. [Answer 2] x 7 = 2.58254417 x 10^{20} that’s cubic millimetres

4. Divide [Answer 3] by 10⁶ for litres and 10⁹ for cubic metres. 2.58 x 10^{14} litres, 2.58 x 10¹¹ m³

5. 2.58 x 10^{10} m² = 2.58 x 10 ⁴ km². So 160.70 km on a side. Wow. Using V = 258.254417 km³.... which seems quite small, perhaps.

6. (258 x 3) ^{1/3} = 9.1845 km. That’s so high, the Y8/9 class reckoned we’d see it from school in Plymouth if we put the pile on top of the Brecons, and they allowed for curvature of the earth. we spent quite a bit of time wondering if the pile could be that shape and stay up, and some more time wondering how to build it (and how to feed the multitudes it would require without eating rice; then we realised that the amount consumed would be very small in comparison — we wouldn’t notice the losses.) At 1km high we can guarantee is stays roughly the same shape and, to calculate the base, we have (2.58x10²x3) ^{1/2} = 27.835 km on a side, about the same size as the Brecons (please check that). So put all the rice on the English plain east of the Brecons and have something bigger than Snowdonia, and a lot bigger still once it has rained.

7. If we made it the pyramid the same height as the base, we would have a pile π r²h / 3 = V....... 2r = 2 (3V/π ) ^{1/3} = 6.27 km = r = h. If the cone is 1km high, r is about 15.7km. d = 2r about 31 km. That’s too big to fit on top of Plymouth, and on top of Cardiff it would be visible from Plymouth, we thought. So “No I can’t” is a good answer.

8. 2.58x10⁵ / 180 = 1.435 km. Same height as the Cairngorms, similar volume. I’m beginning to doubt the numbers again, but they are right. If this pile were spread over Dartmoor, the mean height would be higher than Ben Nevis, which is 1344m high; since most of Dartmoor is over 300m and some of it over 600m, the top would be close to 2000m, allowing for some sagging around the edges. The class spent quite a lot of time checking they hadn’t dropped some zeros. Especially with roots, they discovered you can have a very different looking number — a good side issue to clear up, it turned out. So many kids learned stuff that week without realising it…..

9. Whatever answer 8 was, times 3 to 5. So 4 to 7 km high. But it wouldn’t expand like that, the height would rise a little and the area would spread by a lot. We tried various models for this heap, and in most of them the pile reached the coast and therefore the sea. If it stayed to scale, the increase would be the cube root of the expansion multiplier, more like swapping miles for kilometres. If on Southern Dartmoor, it surely reaches the sea. We could have tried building pyramids in class to discover what would stay up; I moved chairs around to show what was stable and why, but the current fuss about perceptions of safety meant I had more fun than they did. There must be a way around this; a trip to the sand pit?

The density of rice is around 0.9 g / cm³, 0.9 tonnes/m³. World production was 200 million tonnes in 1960 and has been over 700 million tonnes since 2012. 700 million tonnes is 6.3x10^{9} m³, 6.3 km³. Our problem demands around forty years of the current yield, so more than has been grown (and eaten) in the last fifty years. Back in the days of our story, there really was not enough rice, nor had enough rice been grown (in total, ever — now that is scary). To make our story work at all, we would have, overnight, removed a staple crop and turned it into (edible) currency. This is debt on a global scale. Indeed the pile is so extreme I cannot imagine a circumstance in which the debt would ever be paid, not in rice. Far, far cheaper for the emperor to give the young man whatever he wants, in money, a number best expressed as 10^{x}, with x>6, possibly x>10. Giving away the daughter — or whatever is appropriate to the way you rewrote the story— would be an altogether better solution from a national point of view. In the versions of such tales that I read as a teenager, he’d have been bumped off by the grand vizier as soon as the size of the number was appreciated. Hang the consequences, it’s the only affordable solution. Life is cheap, death is cheaper still.

If you decide to use this in class, do NOT let on what might come next. Just let it happen and go with the flow. Get the classd to have ownership of the problem and then you might well be amazed just how much they collectively gain from this. The Y8 class I first did this with took a whole week of lessons—five 40-minute lessons—and, even on the Friday theyu were fired up with extension questions, which is when we found we were discussing things that were more easily labelled as other subjects. It was weeks later they realised how much had been learned, just by actually wanting to know. I wonder how many *still* remember those lessons, at least twenty years later?

DJS Edits, 2020