Sunday, 7 June 2015

The Chessboard And The Rice

When I was about eight or nine...

...I had one of those childhood illnesses - couldn't tell you which one, German measles I think - that can cause fever and delirium. Anyway, I think this is my only really clear memory of being delirious to the point of genuinely believing that something really bad is happening (even though it obviously isn't).

My dad was always into maths as a sort of hobby and one of the maths-related stories he told me was the one about the chessboard and the rice. It's a well-known fable and although details differ between versions, the one I remember goes broadly like this:

There is an Emperor of some empire or other (obviously) and an inventor. The inventor invents something so amazing - possibly the game of chess itself? - that the Emperor wants to give him a huge reward and asks the inventor what he would like. Effectively he is saying "name your price".

But instead of asking for a huge sum of money, or for lands, the inventor says that he would like his reward to take the form of grains of rice, arranged on a chessboard. On square 1 would be 1 grain of rice, square 2 would have 2 grains, square 3 would have 4 grains, square 4 = 8 grains, square 5 has 16 grains, i.e. doubling each time up to the 64th square. The Emperor laughs and asks the inventor why he is requesting such a paltry prize. "No," says the inventor, "this is actually many times more rice than the Empire can produce in a hundred years."

He was of course demonstrating to the Emperor how counter-intuitive a geometric progression can be. The actual amount of grains of rice (r) he would receive (were it possible) would be:

r = 2^0 + 2^1 + 2^2 + 2^3 + ... 2^62 + 2^63, so multiplying both sides by 2:

2r = 2^1 + 2^2 + 2^3 + ... 2^63 + 2^64, or in other words:

2r - r = (2^1 + 2^2 + 2^3 + ... 2^63 + 2^64) - (2^0 + 2^1 + 2^2 + 2^3 + ... 2^62 + 2^63)

r = 2^64 - 2^0

r = (2^64)-1

r = 18,446,744,073,551,615

which is a lot of grains of rice (according to Wikipedia, it would weigh a total of 461,168,602,000 metric tonnes and would make a pile the size of Mount Everest, which seems to assume that there are forty grains of rice to a metric gramme, which sounds about right). Whatever, it's something like a thousand times more rice than is produced in a typical year (these days, not in the times of this Empire - who knows, they might have been really mad on rice and everyone would be cultivating rice all the time - but the principle stands).

It's such a basic principle that it has innumerable real-life applications but that's not why I typed all that stuff about it, even though it was all very interesting and that and suchlike.

Nah, it was because when I had this childhood illness or whatever it was, I have this really clear memory of being in bed, recuperating I suppose, when I was suddenly convinced that the chessboard was real and it was right there and the grains of rice were being counted onto it RIGHT THERE AND THEN. I remember running downstairs in a panic because I had to tell someone that there was this existential threat - everyone was going to be suffocated! - and no-one was doing anything to stop it...

I don't recall what happened after that but on checking, no Mount Everest-sized piles of rice were reported in the northeast of England in the late 1970s or early 1980s, so I think it can be assumed that somebody must have calmed me down and told me it was all a dream. But I perceived it as very real at the time.

It actually ties into one of my genuine fears (I can honestly say I don't have many rational fears) but I've written enough bollocks for now. And anyway, I want to do a whole "thing" about it because it's such a fascinating subject that it deserves its own "thing". But not today.

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