The M1M1 treasure hunt, week 5
Frodo stared in fascination at the golden ring in front of him.
"But what is a 'Ring of Power', Gandalf?" he asked the wizard.
"That, my dear hobbit, is one of the great Polynomial Rings. You may learn about them when you're a little older. They were forged long ago, in the great furnaces of Al Gebra by the Pure Elves, one of the ancient races of MiddleEarth. Each ring provides its bearer with the power to solve equations."
"But who made them, Gandalf?"
"The elflord Cardano, first created 3 rings, each inset with a cubic crystal, that enabled him to unravel ancient third-order mysteries. Then came the Dwarves, ever delving deeper, melting and moulding the very earth, who called for more useful devices, which could be Applied to worldly problems."
"I have heard stories of another great ancient race, the Ents. What of them?"
"Ah, Frodo. The Elementary Number Theorists, or Ents, are lumbering giants who change but slowly with the times - it is said that many still regard real numbers with suspicion. But to return to Ring Theory, the Dark Lord Sauron, heard that many rings did not have "ones" and so created the One Ring, a magic ring of supreme power, which could solve ANY equation, quintics or higher or even other sorts."
"Beggin' your pardon Mr Gandalf," interrupted Sam the eavesdropping gardener, "my ol' Gaffer told me that just can't be done. Not by Elves nor even Wizards, sir."
"Not without magic, Sam. But the One Ring is an evil artefact of great power."
"But even if it does work, wouldn't that be a good thing? We could solve all the problems of the Shire." A distant, crazed look formed on his face. "They'd call me Samwise the Wise, Fields Medallist, Fellow of the Royal Society, I could found my own university - we'd make all the others close down. We'd ban the study of all subjects except Mathematics...our students would work 14 hours a day, seven days a week...no, more..."
"Sam!" cried Frodo, horrified.
"You see, Frodo," observed the wizard calmly, "that is the Evil Ring at work. It amplifies ambition, draws on our dreams and feeds on our fears. If the Dark Lord recaptures it, all is lost. He plans to render even quadratic equations insoluble, maybe even linear equations. Life as we know it would become impossible and devoid of purpose."
"Then what must we do?"
"The ring must be destroyed by casting it into the Pit of Doom! The journey will be long, arduous, very perilous and of the utmost importance. So I'll leave you to it. Take a friend or two with you, don't tell them what's going on and I don't see much point in giving you a map. Good luck."
After some days travel, Frodo slipped the great ring on his finger. A message appeared in his mind.
"This ring is registered to Evil-Dark-Wizards PLC, Mordor. Do not proceed unless you are an authorised Ringbearer. For access, enter the
radius of the ring of convergence of the power series ∑ n! xn."
Frodo's mind suddenly filled with a wondrous clarity. The world around him receded into insignificance apart from the nine perfectly ordinary black horsemen, several miles away but closing fast.
"That was an evil question, Mr Frodo. The sooner that thing is destroyed the better," sulked Sam.
"But now I have access to such wonders, Sam! Polynomials are so easy. I'm in an ideal domain! Limitless knowledge! And knowledge of limits too!
The ratio of x4-2x3+12x2-22x+11 to x3-3x+2 tends to what as x approaches 1, Sam?"
The thundering of hooves echoed down the valley.
In desperation, Sam had made a lucky guess. This madness had to stop -- how could he bring his master back to his senses?
"Mr Frodo, Sir," he began. "I think I saw some writing on the inside of the ring just before you put it on."
Frodo removed the ring from his finger and glanced inside. Sure enough, in large glowing Elven letters was engraved:
"One Ring solves equations all, and leaves no roots behind them."
"One Ring makes Black Riders call - the GPS will find them."
Then in small print at the bottom could be seen:
"To disable Black Rider tracking system enter the limit as x → ∞ of
tanh x + x2e-x + [log(2cosh(2x))/x]3 ."
"Hurry Mr Frodo," wailed Sam, as the fearsome Nazgul dismounted from their steeds and bore down upon the helpless hobbits.
Go to
Week 1
Week 2
Week 3
Week 4
Week 5
Week 6
Week 7
Week 8
Week 9