The way I first heard the story, it happened in ancient Persia. The Grand Vizier (وزير), the principal advisor to the king, had invented a new game. It was played with moving pieces on a square board comprised of 64 red and black squares. The most important piece was the King. The next most important piece was the Grand Vizier. The object of the game was to capture the enemy King. The game was called Shahmat (بازی شطرنج)– Shah for King and Mat for defeat. In Russian it is still called Shakhmat (шахматы). (Even in English there is an echo of this name: Check-Mate). As time passed, the pieces, their moves and the rules of the game all evolved; there is, for example, no longer a Grand Vizier – instead a Queen. And, of course, the game is Chess.
As the story goes, the King was so pleased with this new game that he asked the Grand Vizier to name his own reward. Being such a strategic master, the Grand Vizier had his answer ready.
“I am a modest man, Your Majesty and I wish for a modest reward”, said the Vizier. Gesturing to the eight columns and eight rows of squares on the board he had invented, he asked that he be given a single grain of wheat on the first square, twice that on the second square, twice that on the third, and so on, until each square had its complement of wheat. King was puzzled with such a modest sounding reward and tried to reason Vizier out by offering jewels, palaces and other such riches. Vizier declined all of that. King, marvelling at the humility and restraint of his counsellor, consented.
However, when the master of the Royal Granary started to fill up the squares, starting with the small number of 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024...king was in for an unpleasant surprise by the time the 64th square approached. The numbers of grains become colossal, staggering. Whether the King handed his entire Kingdom to the Vizier or started taking lessons in geometric progression, no one knows what happened next.
To appreciate King’s dilemma, let’s see the exact calculation using exponents:
The exponent just tells us how many times we multiply 2 by itself.
22=4. 24=16. 210=1,024, and so on.
Call G the total number of grains on the chessboard, from 1 in the first square to 263 in the 64th square.
This means G = 1+2+22+23+...+262+263
By doubling both sides of the above expression, we find
2G =2+22+23+...+263+264
Subtracting the first expression from the second gives us
2G –G = G = 264-1, which is the exact answer.
How much is it roughly in ordinary base-10 notation?
210 is close to 1,000 or 103 (within 2.4 percent)
And, 220 = 2(10x2) = (210)2 = roughly (103)2 = 106, which is 10 multiplied by itself 6 times or a million.
Likewise, 260 = (210)6 = roughly (103)6 = 1018
So, 264 = 24 x 260 = roughly 16 x 1018
Or, 16 followed by 18 zeroes, which is 16 quintillion grains.
(A more accurate calculation gives the answer 18.6 quintillion grains)
How much does 18.6 quintillion grains of wheat weigh?
If each grain is a millimetre in size, then all the grains together would weigh around 75 billion metric tons, which far exceeds the Shah’s wheat storage. In fact, this is the equivalent of about 150 years of the world’s present wheat production.
Had chess been invented with 100 (10 x 10) squares instead of 64 (8 x 8), the resulting debt in grains of wheat would have weighted as much as the earth.
Check-mate!
Source: Billions and Billions, by Carl Sagan
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