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Sunday, August 9, 2015

The Amazing Chessboard Theory

The Amazing Chessboard Theory
  
The wheat and chess problem appears in different stories about the invention of chess. One of them includes the geometric progression problem. Its earliest written record is contained in the Shahnameh, an epic poem written by the Persian poet Ferdowsi between c. 977 and 1010 CE. Another version has the inventor of chess (in some tellings Sessa, an ancient Indian Brahmin) request his ruler give him wheat according to the wheat and chessboard problem. The ruler laughs it off as a meager prize for a brilliant invention, only to have court treasurers report the unexpectedly huge number of wheat grains would outstrip the ruler's resources. Versions differ as to whether the inventor becomes a high-ranking advisor or is executed. 

The Original Story:The Full story reads as follows:

Chess is one of the world’s most popular games. The Iranians and Indians might be their inventors based on ancient scrolls which have been found in both territories. On the one hand, chess was known in Iran as shatranj, on the other hand chess was known as chaturanga in India. Both shatranj and chaturanga are very similar to the current chess. Their pieces symbolized the four forces of their armies: infantry, cavalry, elephants, and carriages. There are a lot of stories about chess. According to the Indians there was a famous philosopher who lived on a high mountain. His power was very amazing and he was able to create this complex game in a night of magical inspiration. 

  
Legend has it that in the province of Taligana, in India, lived for many years a rich and generous king named Iadava. An adventurer named Varangul attacked Iadava’s kingdom. He had to wield his sword, and in front of his army, faced Varangul’s army. Iadava, who had a military genius, defeated Varangul in the fields of Decsina, but he paid a heavy price for his victory, his son Adjamir died in combat. There was so much sadness in Iadava’s heart that he locked himself in his castle, and no longer wanted to talk more with anyone. His only consolation was to repeat the maneuvers of combat in a sandbox, as a tribute to the memory of beloved son Adjamir.

But one day it came to the sad palace, a young Brahmin named Lahur Sessa, from the village of Manir. He asked the guards to see the king saying that he had invented a game especially for him in order to cheer his hours of solitude. Iadava decided to receive Lahur Sessa in his palace. He had a great curiosity to see the game which had been invented for him. When Lahur Sessa was in front of the king, he gave him a beautiful board divided into sixty-four squares and thirty-two pieces: sixteen of white and sixteen of black. Each group of pieces represented according to Lahur Sessa, two armies, the army of Varangul and the army of the king. After some brief explanations, the king began to play with great enthusiasm, really was fascinated with the new game. And it happened that Iadava had to sacrifice a rook to win the game (changing one more valuable piece by other one less valuable). This opportunity was exploited wisely by Lahur Sessa. He told the king, "Sometimes we need to make a sacrifice to achieve a greater good for everyone."

Iadava caught the acute observation which made a reference to his son Adjamir, sadly died in combat. Pleased with the beautiful game that Lahur Sessa had invented for him. Iadava told Lahur Sessa, "Ask me what you want to and I will give you immediately." Sessa kindly explained to Iadava, "I would like to receive a grain of wheat for the first square, two grains of wheat for the second square, four grains of wheat for the third, eight grains of wheat for the fourth, and so on until the sixty-fourth square." By hearing such a humble request Iadava began to laugh nonstop. After a while, he ordered that he would be given what he had requested. Later mathematicians confusedly came to the king to tell him that it was impossible to accommodate that request. The quantity of wheat was so great that all wheat from his kingdom was not enough wheat to pay what he had promised Lahur Sessa. This is the incredible account of wheat after reaching the sixty-fourth square: 18,446,744,073,709,551,615.
Iadava was amazed by such an impressive figure. He told Sessa, "Unhappy is he who assumes the burden of a debt whose worth cannot be measured by the simple means of his own intelligence."Iadava embraced Sessa and appointed him to first vizier for life.
There are conflicting versions of the above story , but suffice to say it has achieved its intention of giving the message in it so correctly.

The wheat and chessboard problem (sometimes expressed in terms of rice instead of wheat) is a mathematical problem in the form of a word problem. Consider it terms of even one $ or one Rupee on each chess board square and see what it shows at the end of this calculation!!!!
If a chessboard were to have wheat placed upon each square such that one grain were placed on the first square, two on the second, four on the third, and so on (doubling the number of grains on each subsequent square), how many grains of wheat would be on the chessboard at the finish?
The problem may be solved using simple addition. With 64 squares on a chessboard, if the number of grains doubles on successive squares, then the sum of grains on all 64 squares is: 1 + 2 + 4 + 8... and so forth for the 64 squares. The total number of grains equals 18,446,744,073,709,551,615, much higher than what most intuitively expect.
On the entire chessboard there would be 264 − 1 = 18,446,744,073,709,551,615 grains of wheat, weighing 461,168,602,000 metric tons, which would be a heap of rice larger than Mount Everest. This is around 1,000 times the global production of rice in 2010 (464,000,000 metric tons)
Consider this even if you were to take ask someone to place 1$ or 1Rupee and give it to you what would happen - =$18,446,744,073,709,551,615 or Rs. 18,446,744,073,709,551,615? That is 18 quintillion 446 quadrillion 744 trillion 73 billion 709 million 551 thousand and 615 in whatever currency you count. No two doubts about what would have happened to the person who requested the King to give that much wheat.
There is also a Mathematical Theory explained which can be seen below and also read in the Wikipedia Link:


The Moral of this Theory/Story
On a lighter note but can be taken quite seriously if taken in the true sense the chess board theory explains a lot of hidden principles and words of wisdom that few read through or even try to analyze and understand.
The problem warns of the dangers of treating large but finite resources as infinite, i.e., of ignoring distant but absolute and inevitable constraints. As Carl Sagan wrote when referencing the fable, "Exponentials can't go on forever, because they will gobble up everything. Similarly, The Limits to Growth uses the story to present the unintended consequences of exponential growth: "Exponential growth never can go on very long in a finite space with finite resources.

Courtesy: Information collated from : http://www.chess.com/news/a-beautiful-story-of-chess-1078

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