We have all read or heard the story of the Chinese emperor and the wise man who had helped him build his great empire. He wanted to reward the man for his services and asked him to name his reward. The man brought out a chessboard and asked the emperor for enough rice such that starting with one grain of rice for the first square of the board, the next would have two, and each succeeding square will have the number of grain multiplied by two, i.e. four, eight, sixteen, and so on. The emperor laughed and said that would be easy. When it came to meeting the request the emperors granary became empty, he went broke, and the easy became impossible.
Let us see the numbers involved. The chessboard has sixty-four squares. Starting from one grain for the first square the number of grains for the last square would be two raised to a power of sixty-three. The total number of grains to meet the requirement would be the sum of the geometrical series starting from one and ending w ith two raised to the power of sixty-three. This is a colossal number and we have no word to describe it, so let us use some approximations. Two raised to power ten is one thousand and twenty-four. Let us round it out and say it is a thousand. If we were counting money, twenty-four would still be significant but for our purpose it is not that important. After all it is done in Computers all the time when we use the term kilobyte.
Continuing with the numbers in the series the twenty-first term would be two raised to power twenty, that is a million (in order of magnitude). The thirty-first term would be a billion, forty-first a trillion, and fifty-first a quadrillion. We still have thirteen more terms to go but let us stop here because we do not have words for those numbers. Million and billion we a re quite familiar with and we are just getting to have an idea about trillion with our national debt, but quadrillion is still vague. To put it in perspective we have only to remember that it is a sixteen-digit number - one followed by fifteen zeros. When we get to term sixty-one we add three more zeros. Taking the remaining three terms and adding them all we add another zero. So the total number of rice grains for the reward is (in order of magnitude) one followed by nineteen zeros.
Do we have an idea how much will this rice weigh? Let us assume that one kilogram of rice may contain twenty thousand grains. One metric ton is one thousand kilos. So the number of rice grains contained in one ton will be two followed by seven zeros. The weight of the rice needed for the request will roughly be five followed by eight zeros tons, which is five hundred million tons. This is a colossal amount well beyond the resources of even a great emperor. However, if the emperor knew anyt hing about geometric progression, he would have responded to the request differently. He could have granted it with the condition that the wise man himself does the counting.
Let us assume that a person can count ten grains per second. The number of seconds in one year is approximately thirty one million, an eight-digit number. Even taking just an order of magnitude estimate it would take almost over thirty billion years for one person to count all the rice grains. The age of the universe is estimated to be around fifteen billion years.
Dharmbir Rai Sharma is a retired professor with electrical engineering and physics background. He obtained his M.S. degree in physics in India and Ph.D. in electrical engineering at Cornell University. He has taught in universities here and also in Brazil, where he spent sometime. He maintains a website http://www.cosmosebooks.com devoted mainly to philosophy and science.
Author:: Dharmbir Sharma
Keywords:: Geometric p rogression, Power of two, Chinese emperor and wise man
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