Chess, from a fountain of genius

Dedicated to Captain Stephen and the Staines Chess Club Newsletter by David Wilson
OBEOBE = On Board Expert

See comments to this issue in the bottom

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.

The bored - The size of the chessboard on which the grains were to be placed, would have to be 26½km (16½ miles) square.  The ceiling height, (indoors in case of rain) being 3½km.
Proof - Assuming all the squares of the chessboard to be the same size, and that the grain dropped centrally created the shape of an inverted ice cream cone of equal base length to height (using averaged constants for mass, shape, humidity, gravity, friction and colour).  The last square having the most to hold (9,000,000,000m3), had sides roughly equal to the diameter of the inverted ice cream cone.  Hence the volume of rice, 9,000,000,000m3 was equal to the volume of a cone, shown as; 1/3 x 22/7 x r2 x h.  Where; 22/7 is Pi (I couldn’t find the 16th letter of the Greek alphabet on my keyboard, or any of the others for that matter), which denotes the ratio of a circles circumference to diameter, r = radius of the cone’s base, h = height of cone (in this case equal to the cone’s diameter = 2r).  Thus 9,000,000,000m3 = 1/3 x 22/7 x r2 x 2r (= 44/21r3).  Hence r = 1650m.  Therefore the diameter of the cone = 2r = 3.3km.  Therefore the board length (8 squares) = 8 x (3.3km + ALBM) = 26½km.  Where ALBM = a little bit more (so one can see the edges of the fullest square).

Incidentally, the floorboards under the chessboard would have had to have been supported by joists of greater thickness than any tree in existence.

Transportation - If each grain of rice had an average volume (very conservatively) of one cubic millimetre, they would collectively form an 18 kilometre cube.  That is, 50 million trailer-truck loads or, to put it in context and assuming a 500g bag of Tesco value rice occupies a space of 120x70x50mm hence a 0.84ton(1m3) payload each …18 thousand million camels!  Which are probably more camels than have ever existed.

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!

Agriculture - Should he wish to grow the grain, at a 5 second per 4cm3 handful per M2 of land ‘sow rate’.  It would require an army of 10,000 peasants working 20 hours a day, 7 days a week, 52 weeks a year for 5,365 years, 199 days, 12 hours, 28 minutes, and 12 seconds.  Presumably they are still scattering somewhere.

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!

Loaves and fishes - If he wished to cook the rice (Basmati assumed).  With generous 75g helpings per person and an average 3kg post-dressed chicken yielding 4 portions of breast meat, the legs and wings being sort of frozen until Colonel Saunders could find a use for them later.  It would take a chain of 10,000 ‘Vindaloo Nites’ restaurants selling 1000 Chicken Biryanis a day, 7 days a week, 52 weeks a year…a 56 Delia Delirious thousand years, allowing for Bank Holidays, to chomp through this ‘tekaway’.  Christ!  I wonder if they used chopsticks?!  Not to mention (assuming a ‘living’ cage 300x200x200mm per unit, stacked in 2m columns with 1m wide access ‘seeding and weeding’ aisles) the 10,000 10kmx5km chicken battery factories!!!...or… to put it another way, assuming average life expectancy of 25years (well, no medical insurance you see) and the rather optimistic luxury of 3 meals a day.  He could feed his wife and kids, Mum, Dad, weird Uncle Bob, the screaming urchins of his village, their flea bitten parents and the rest of the estimated 300 million population of earth at that time and their next 25 generations to come, allowing for the ancient festival of Karwa Chauth, the fast for married women, tradition is a wonderful thing!
In conclusion - Maybe this chap wasn’t as shrewd as our teacher thought.  Either that, or he took an IOU that not even the Mafia could have enforced or he was a very hungry boy indeed or had large trouser pockets or 28 million, million pairs of normal trousers with front and back pockets stuffed full of rice.  QED.

Comments

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:

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
Hope you're well,  Richard.....(Well I was..................DOH!)  
I think David is closer by a lucky double mistake. Looking at the next table will help:
SquarePower of 2Grains for the squareAccumulated Grains
120 = 11
221 = 23
322 = 47
423 = 815
n2n-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