About 1260 AD, Ibn Khallikan, a Kurdish historian livingin The Abbasid Empire (modern Iraq), wrote an encyclopediawith biographies of many famous men (though no women). Oneof the biographies includes a story about chess and the meaning of "exponential growth".

The story takes place in India, because IbnKhallikan knew that chess was a game that came from India. According to this story, King Shihram was a tyrant whooppressed his subjects. One of his subjects, a wise man namedSissa ibn Dahir, invented the game of chess for the king to play, to show him that a king needed all hissubjects and should take good care of them. King Shihram was so pleased that he ordered that the gameof chess should be preserved in the temples, and said that it was the best thing he knew of to traingenerals in the art of war, a glory to religion and the world, and the foundation of all justice.

Then King Shihram asked Sissa ben Dahir what reward he wanted. Sissa answered that hedidn't want any reward, but the king insisted. Finally Sissa said that he would take this reward: the kingshould put one grain of wheat on the first square of a chessboard, two grains of wheat on the secondsquare, four grains on the third square, eight grains on the fourth square, and so on, doubling thenumber of grains of wheat with each square (an exponential rate of growth).

"What a dummy!" thought the king. "That's a tiny reward; I would have given him much more."He ordered his slaves to bring out the chessboard and they started putting on the wheat. Everythingwent well for a while, but the king was surprised to see that by the time they got halfway through thechessboard the 32nd square required more than four billion grains of wheat, or about 100,000 kilos of wheat. Now Sissa didn't seem so stupid anymore.

Even so, King Shihram was willing to pay up.But as the slaves began on the second half of the chessboard, King Shihram gradually realizedthat he couldn't pay that much wheat - in fact, to finish the chessboard you would need as much wheatas six times the weight of all the living things on Earth.


a.- According to the text, could you confirm the amount of grains of wheat there are on the 32nd square?
b.- Could you calculate the number of grains it should have on the 64th square?