Thursday, June 13, 2013

Source of the Cipher 4: Computer's Delight

            4. Computer's Delight

            “And so you ran for me,” said Ramanujan. “Prince Rahni’s arithmetic clerk.”
            “My husband, the lightning calculator,” said Namagiri.
            They snuggled closer together. Ramanujan whispered into his wife’s ear, “Shall I tell you what they asked and how I answered?”
            Namagiri smiled, but that was invisible in the darkness, so she said, “Yes.”
                        *                      *                      *
            Ramanujan bowed deeply. “O Radiance! Your Excellency! Enlightened Star of India!” He bowed deeply again, as protocol required. “O Victor, Lawgiver and Sage! Worthy Inheritor! Noble Founder!” He bowed a third time. “Sri Sri Magnificence Sri Sri Lucidity Sri Sri Great-Souled Sri Sri Sri Rahnraygunaurobindo! I await your request!”
            Prince Rahni said, “What our guest the Sheik and I want from you, Ramujana.... Rajanuma... Ranamumu... what was your name, again?”
            “Ramanujan, O Radiance! Your Excellency! Enlightened Star -”
            “ - no need for formalities,” Prince Rahni said kindly. “Just plain ‘sire’ will do. Today I ask you to demonstrate your mathematical talents for our esteemed guest. You are to do this by the rapid computation of solutions to sundry mathematical problems.”
            The Sheik asked, “These problems; are they to be proposed by me or by you?”
            Prince Rahni said, “Why, let us alternate; and you may start.”

            The Anise Seeds
            Sheik Kahmunni nodded his head, chuckled, and said, “Then consider this... mathematical problem.” He reached into the bowl of anise seeds; he took out a handful of seeds; and he flung that handful across the banquet room floor.
            “How many seeds is that?” he asked Ramanujan.
            Ramanujan scanned the floor; he looked up and down, left and right. He gazed blankly into the air, mumbled to himself, scratched his head, then announced, “It is the smallest number that can be written as a sum of two cubes in two different ways.”
                        *                      *                      *
            “ - and they just stared at me,” Ramanujan complained.
            Namagiri smiled in the darkness. “Now why would they want to do that?”
            “I have no idea,” Ramanujan declared. “My answer was simple and obvious.”
            “No doubt it is to you, my dear; but perhaps not to them.”
            “So I deduced, after a few strained moments. I patiently explained that the number was one thousand, seven hundred and twenty nine. But still they did not understand.”
            Namagiri asked, “And when they persisted in not getting your little joke, were you forced to explain it?”
            “Alas, I was. I actually had to tell them that one thousand seven hundred and twenty nine equals one cubed plus twelve cubed; and also nine cubed plus ten cubed; and that it is indeed the smallest number that can thus be written as a sum of two cubes in two different ways.”
            Namagiri said, “One plus twelve dozen dozen.”
            Ramanujan said, “Yes.”
            Namagiri said, “Which was how many anise seeds were scattered across the floor.”
            “Why... yes, of course.”
            Namagiri continued, “A number that you were able to count by scanning the floor, gazing into the air, mumbling to yourself, and scratching your head.”
            “Why... yes, of course.”
            “And funniest of all,” Namagiri continued, “that is what Prince and Sheik found most impressive. Not your instant sums of cubes, but your lightning counting.”
            Ramanujan sighed deeply. “Just so, dear heart.”
            “It’s expectable,” Namagiri consoled him. “We understand them, all too well; but when will they ever understand us?”

                        *                      *                      *
            The Maldives problem
            “My turn,” Prince Rahni told the Sheik; then he said to Ramanujan, “I have a problem for you; a rhyming riddle.”
            The Sheik said, “Then by all means, please recite it. No, do not hesitate on my behalf; were you to sing well, then we shall all be pleased, and were you not to sing well, then rest assured that I myself have sung worse.”
            Prince Rahni smiled ingratiatingly at his guest and said:
            “As I was sailing through the Maldives
            I met a man with seven wives;
            and every wife had seven sons,
            and every son had seven boats,
            and every boat had seven goats,
            and every goat had seven packs,
            and every pack had seven sacks,
            and every sack had seven cats,
            and every cat had seven kits.
            Kits, cats, packs, sacks, goats, boats, sons, and wives;
            How many were sailing through the Maldives?”
            Ramanujan responded by rolling his eyes, clicking his tongue, mumbling to himself, and exhibiting various other tics that Prince and Sheik scrutinised, fascinated; then Ramanujan said, “There were:
            seven wives;
            forty-nine sons;
            three hundred and forty three boats;
            two thousand, four hundred and one goats;
            sixteen thousand, eight hundred and seven packs;
            one hundred and seventeen thousand, six hundred and forty nine sacks;
            eight hundred and twenty-three thousand, five hundred and forty three cats;
            five million, seven hundred and sixty-four thousand, eight hundred and one kits;
                        making a total of:
            six million, seven hundred and twenty five thousand, and six hundred.”
                        *                      *                      *
            “Wrong!” Namagiri exclaimed.
            “Wrong, you say? How odd,” Ramanujan mused.
            None of them were sailing through the Maldives!”
            “None, you say? Odder still.”
            “In fact, I was sailing through the Maldives!”
            “You were, darling? Oddest of all!”
            “No, no, not me personally,” Namagiri said, laughing. “Silly computer, didn’t you hear how the riddle began? ‘As I was sailing through the Maldives...’ The narrator, not that whole village!”
            Ramanujan said, “How very odd indeed! For what you have just told me is just what Prince Rahni told me. He too said ‘Wrong!’, he too said none were sailing through the Maldives, he too said that he was sailing through the Maldives, and he too laughed at me, then cited the riddle’s first line. He even used the phrase ‘the narrator, not that whole village’.”
            Namagiri said, “It takes a village to support five million kits.”
            “Five million, seven hundred and sixty-four thousand, eight hundred and one,” Ramanujan corrected her, then continued, “And so I must tell you what I told my Prince; that I reluctantly concede the logic of your analysis.”
            “Poor Ramanujan! Are your feelings hurt by being caught in the riddle’s trap?”
            “No, I am merely puzzled. How did he know what you were going to say?”
                        *                      *                      *
            Divisions of Time
            “And now my turn,” said the Sheik. He turned to Prince Rahni. “Are you familiar with the science of the stars?”
            “Astrology? I swear by it,” said Prince Rahni. “As does the Lady Nan-See. We make all our major military and financial decisions with its help.”
            “How clever of you,” the Sheik said smoothly. “Astrology is a most exacting science, at least in its observational phase. The subtlety of Heaven’s rhythms have forced certain astrologers I know to invent new units of time. Computer, attend; they define an ‘hour’ as one part in twenty-four of a day; a ‘minute’ as one part in sixty of an hour, and a ‘second’ as one part in sixty of a minute.”
            “An ‘hour’ is one twenty-fourth of a day, a ‘minute’ is one sixtieth of an hour, and a ‘second’ is one sixtieth of a minute,” Ramanujan repeated obediently.
            “Astonishing rigor!” Prince Rahni marvelled. He shook his head. “To think that human beings should find it necessary to worry about such brief spans of time!”
            “I have no doubt that Your Highness never allows the passage of time to bother him in the slightest,” Sheik Kahmunni said smoothly. “Computer, tell me; how many seconds are in a day?”
            Ramanujan instantly replied, “There are eighty-six thousand, four hundred.” 
            “How many seconds in a year?”
            “In a year of three hundred and sixty-five days, there are thirty-one million, five hundred and thirty-six thousand seconds.”
            “How long is a thousand seconds?”
            “Sixteen minutes and forty seconds.”
            “How long is a million seconds?”
            “Eleven days, thirteen hours, forty-six minutes, and forty seconds.” 
            “How long is a thousand million seconds?”
            “About thirty-one years, eight months, and eight days, give or take a few days, depending on leap years and lengths of months.”
            “And how many seconds long is a hundred years?”
            “Averaging for leap years, we get about three thousand million, one hundred and fifty five million, seven hundred and sixty thousand seconds.”
            “Astonishing speed,” said the Sheik. “I conclude my turn.”

            The Chariots and the Scarab
            Prince Rhani said, “Once upon a time, many cosmic cycles ago, the god Shiva and the goddess Kali were angry at each other; so they boarded war chariots. These chariots faced each other across a vast plain; the domain of Lord Indra. When the trumpet sounded, the two war chariots hurtled directly towards each other at breakneck speeds. Both Shiva and Kali vowed to prove the other a coward by not being the one to swerve first.”
            “Aha,” said Sheik Kahmunni, “a contest of resolve! I have heard of young men back home in Arabia attempting such escapades; they’d ride horses at each other, taunting the other to veer aside. Alas, the horses they ride never co-operate with such games.”
            “But these were magic chariots, responsive to every whim of their riders.”
            “Even if that whim risks a fatal collision?”
            “Yes indeed.”
            “Well! That explains everything. Horses have much more sense,” said Sheik Kahmunni. “But please, continue your story.”
            Prince Rahni continued, “Lord Vishnu, desiring peace, attempted to mediate between the quarrelling parties. To this end Vishnu incarnated himself as a scarab beetle; and in this form he flew from Shiva to Kali to Shiva to Kali to Shiva to Kali; bouncing back and forth between the chariots, shuttling faster and faster as collision neared.
            “Assume that Lord Indra’s domain was a million leagues across. Assume also that each chariot moved at a rate of ten thousand leagues in an hour. Assume moreover that the Vishnu-scarab started his peace mission from the prow of Lord Shiva’s chariot, that he flew at a rate of a hundred thousand leagues per hour, and that he did not cease his accelerating shuttling, even unto the moment of collision itself.
            “Assuming all this,” said Prince Rhani. “how far did Vishnu travel before the chariots collided and the universe was destroyed?”
            The sheik exclaimed, “Destroyed?!”
            Prince Rahni shrugged. “It was a long time ago. Don't worry about it...”
                        *                      *                      *
            Namagiri asked, “And did you give an answer?”
            “Yes, dear, and promptly. Lord Vishnu flew a distance of precisely...”
            “... five million leagues,” Namagiri interrupted.
            “Why, that is just so!”
            “I used the shortcut,” Namagiri said modestly. “No doubt you did too.”
            Ramanujan said, “How odd. Prince Rahni also mentioned a shortcut, which he believed I used. No doubt he meant the infinite series summation.”
            “Why... no. What is an ‘infinite series summation’?”
            “What is your shortcut?”
            “Let us hear your method first.”
            “Why, Namagiri, it is simplicity itself! When the Prince finished telling me the problem, I asked myself; what form is the length of Lord Vishnu’s flight? The answer was obvious; it is an sum of infinitely many terms, decreasing by a constant proportion; a geometric series!”
            “If you say so, dear husband,” Namagiri said.
            “I could explain it more fully...”
            “No, no, do go on,” Namagiri said hastily, for she had heard some of Ramanujan’s ‘full explanations’ before.
            Ramanujan said, “So I asked myself, what is the first term? A moment’s thought revealed that it was ten million divided by eleven. And what is the constant ratio?”
            “No doubt it was crystal clear to you!”
            Oblivious to her hint, Ramanujan barrelled on. “It was obviously nine-elevenths.”
            “And from this what follows?”
            “Why, that Vishnu’s flight had a length equal to this infinite geometric series:
            Ten million divided by eleven
            Ten million times nine,
                        divided by eleven times eleven
            Ten million times nine times nine,
                        divided by eleven times eleven times eleven
            Ten million times nine times nine time nine,
                        divided by eleven times eleven times eleven times eleven
                                    and so on for infinitely many terms.”
            “But of course,” Namagiri murmured. “And from this...?”
            “But it is so obvious! But not to you, my dear? Oh... Well, such a series will sum to a number equal to the first  term divided by one minus the common ratio. One minus the common ratio equals one minus nine-elevenths; that is, two elevenths. Ten million divided by eleven, itself divided by two elevenths, equals ten million divided by two, or five million. Therefore Lord Vishnu flew five million leagues before being destroyed, along with the rest of the universe.”
            “It was a long time ago,” Namagiri said.
            “I’m not worrying about it,” Ramanujan said. “But I am worrying about that shortcut. You used it, Prince Rahni said I must have used it; but what is it?”
            “Well, consider the matter thus,” said Namagiri. “You said that the chariots started a million leagues apart, did you not?”
            Ramanujan said, “Yes I did, beloved.”
            “And that each chariot ran at a rate of ten thousand leagues in a hour?”
            “That is indeed so.”
            “And therefore these chariots approached each other at what rate?”
            “At a rate of twenty thousand leagues per hour.”
            “And therefore how long was it before they were doomed to collide?”
            Ramanujan said, “In a number of hours equal to one million divided by twenty thousand; that is, in fifty hours.”
            Namagiri nodded gravely. Her chin dug into his shoulder. “So I too calculated. And how fast did you say the Vishnu-scarab flew during those fifty hours?”
            “At a rate of a hundred thousand leagues per hour.”
            “And how far would Vishnu fly, given that he flew at a hundred thousand leagues per hour for fifty hours?”
            Ramaujan hesitated, then said, “Oh. He would fly a total distance of a hundred thousand times fifty; that is, a distance of five million leagues.”
            Namagiri nodded gravely. “So I too calculated. How nice to see that your calculation matches mine; for it suggests that we may both be right.”
            “But your method is so much simpler!” Ramanujan exclaimed. He sighed hugely, then said ruefully, “That is the difference between us, my dear. You are a mathematician, but I am just a computer. If only the Sheik understood that!”
                        *                      *                      *
            “A most impressive demonstration,” the Sheik said. “How do you do it?”
            “How do I do what?”
            “The counting, the adding, the lighting multiplication. Tell me, computer; how?”
            Ramanujan blurted, “Unable to comply.”
            The Sheik growled, “And why not?”
            Ramanujan hastily said, “No, no, nothing personal sir... it’s not that I would not tell you; I simply cannot tell you. My skill with numbers came to me when I attained manhood; it is a feel for form and quantity that I cannot explain, only exercise. I wish I could explain how I do these simple tricks; but teaching these simple tricks to others is a trick that I have not yet mastered.”
            Sheik Kahmunni frowned, stroked his beard. “Not good enough...”
                        *                      *                      *
            Ramanujan’s Dream
            “Poor Sheik! Doesn’t he trust your figures?”
            “I doubt he trusts his own,” said Ramanujan. “But how could I explain to that fierce, lean wanderer that you, my wife, gave me my skills?”
            “Oh, you flatterer! And how did I do that?”
            “You inspired me. Have I not told you my dream of you?”
            “Yes,  you have,” said Namagiri, “Often. But please tell it to me again.”
            Ramanujan said, “Once, when I was a child, I dreamed of another life I had. I was far away from here and now, both in space and in time. I was in a distant century in a far land; a cold, rainy land of magicians. Their arts were subtle, perilous and powerful. It was a strange place... but somehow, it was yours.”
            “Yours, Namagiri. Do not ask me how; I cannot explain. In my dream I was alone; but every night (in my dream) I would dream (in my dream) that you would come to me.”
            “A dream within a dream! You were all wound up like a snake biting its tail!”
            “In those dreams-within-a-dream you would come to teach me mathematics. And those visions were all I ever saw of you, in that life.”
            “Tell me, husband, what sort of mathematics I would teach you.”
            “Shall I tell you something complicated, or something a bit simpler?”
            “How about something very very simple? Let’s say, the very simplest thing I ever taught you in a dream within a dream?”
            And Ramanujan said,

            “One-half plus half the square root of five
            One plus the reciprocal of
                        one plus the reciprocal of
                                    one plus the reciprocal of
                                                one plus the reciprocal of
                                                            ... and so on forever!”

            Silence in the darkness.
            After awhile Ramanujan said, “Perhaps I should write it out...”
            “But you know I cannot read.”
            “It is a simple formula; no words, only numbers.”
            “But it is pitch black! What will you write on?”
            “I’m sure we can find something... I have it. Roll over, dear. Yes, onto your back...” Ramanujan caressed Namagiri’s stomach. “Good. Now listen. One-half plus half the square root of five - ”
            “- what a strange number!”
            “It is what geometers call the Golden Mean. It is also equal to the following expression...”
            Ramanujan moistened the tip of his right index finger; then with it he wrote the following across Namagiri’s bare belly:

       1 +      1     
            1 +      1      
                  1 +      1      
                        1 +      1     
                              .  .  .

            Silence in the darkness.
            After awhile Namagiri said, “O gorgeous brilliant romantic husband, this mathematics must be yours. It is far too pretty to be mine!”

No comments:

Post a Comment