Many years ago, when I was at Junior School, a teacher imparted an interesting ‘fact’. Our class were told that the inventor of the game chess was brought before his king and offered any gift he desired. Lamborghinis and Ferraris being a bit thin on the ground two thousand years ago, he ‘shrewdly’ requested, one grain of rice on the first square of his chessboard, two grains on the next square, 4 on the next, doubling all the way up to the 64th and final square.
Our teacher then asked us how many grains that would have been. We all ‘clattered’ out the largest numbers we could think of, mine being in the high hundreds. The class bookworm, who read even during lessons, (how right he was to do so) momentarily, unburied his nose to intimate his suspicions that the figure may in fact be in the region of 50,000 or more. The rest of the class erupted in scoffs, that were only interrupted by the teachers affirmation that his had been the most accurate answer and that is his own estimation (which wasn’t worth the paper it was written on) it would have made him one of the richest farmers in the country…Is that so Mr Fountain of knowledge,…little did he realise…
Using my Nokia phone calculator, and therefore probably rather inaccurately, not that that is going to stop me, I have come up with some interesting conclusions with regards the alleged incident.
The number of grains was about 18 trillion (ie 18 million, million, million) or, if you prefer, 18,000,000,000,000,000,000, give or take a bushel.
Incidentally, the floorboards under the chessboard would have had to have been supported by joists of greater thickness than any tree in existence.
Making like Dr Dolittle, he would have had to call upon the assistance of elephants, hippos, kangaroos, turkeys, humming birds, ants, pet tortoises, hamsters, dingoes, South American (not that it had been discovered yet depending on your point of view) three-toed sloths and every other single living organism on the planet just to shift the stuff!
These 281 million, million handfuls would cover 281 million km2 of land which is nearly twice the entire land surface area of the Earth. Ploughing up the moon would get rid of a bit more, but think of the petrol costs!
This is the response from one of my pupils, Richard, just before he went off to read Maths at Cambridge....
I've read the article you sent me, and checked the calculations as asked and have come up with some of my own figures:Hope you're well, Richard.....(Well I was..................DOH!)
He would have about 37000000000000000000 grains of rice - about double the figure in the article.
This would weigh about 1.2E12 (E12 = times ten to the power 12) tonnes - 1.2 Gigatonnes, and is equivalent to about 2.4E15 bags of tesco's 500g rice. If Mr. Tesco could sell all this rice he would make about 6E14 pounds sterling - 600000 billion pounds. About 20000 times Bill Gates current fortune.
By my calculations the amount of rice on the last square would be twice the amount you have quoted, and would have a volume of 6.4E11 cubic metres. This gives a value of r by your calculation of 6700m, therefore board length would be 1080km.
Anyway, that's quite enough rice calculations for me (the only actual mistake you made was you didn't add up all the rice on all the squares, you simply calculated the amount on the final square), and I will try to remain in touch,Richard
I think David is closer by a lucky double mistake. Looking at the next table will help:
Square Power of 2 Grains for the square Accumulated Grains 1 20 = 1 1 2 21 = 2 3 3 22 = 4 7 4 23 = 8 15 n 2n-1 = ... 2n-1
Then for the square 64, the amount of corresponding grains would be 263 not the power 64 as David assumed, but the accumulation of grains would be by then 264-1 = 18 446 744 073 709 551 615.Alfredo