Exploring Technology in a Wizard World-Chapter 284 - 283 The Poincaré Conjecture and Lickerel
Chapter 284: Chapter 283: The Poincaré Conjecture and Lickerel Numbers
Chapter 284 -283: The Poincaré Conjecture and Lickerel Numbers
“How do you see it?” Great Scholar Socrates asked, looking at Richard.
Richard’s gaze moved away from the title on the papyrus scroll, and with a sparkle in his eyes, he said, “22 days.”
“Huh?” Great Scholar Socrates was startled. “What 22 days?”
“If you solve the problem the right way—finding the impostor scholar Sula’s thief Rade hidden in the secret room—it would take a maximum of 22 days,” Richard said.
Socrates stared at Richard for several long seconds before contemplating, then he nodded his head appreciatively, “Mhm, not bad, it aligns with one of my previous guesses, exactly, 22 days. Now, lad, explain your thought process, let’s see if you’ve got anything different, or wrong, compared to mine.”
“It’s possible to approach the problem by numbering all thirteen rooms—from number 1 to number 13. In the scenario presented in the question, the thief Rade changes rooms, either from even to odd—for instance, from room 2 to room 1, or from odd to even—from room 1 to room 2.”
“With that in mind, we make two hypotheses: on the first day, the thief Rade is in an even-numbered room; or alternatively, on the first day, the thief Rade is in an odd-numbered room.”
“If the thief Rade is in an even-numbered room on the first day, then on the first day we check room 2, the second day we check room 3, the third day room 4, continuing until we search room 12 on the eleventh day. During this process, there’s a very high likelihood of finding the thief Rade. Because the distance between the impostor scholar Sula and the thief Rade will definitely be even—either 0 or a multiple of 2. When the distance is 0, it means the search has been successful, and the thief Rade is caught.”
“If, after this search, the thief has not been found, that will imply that on the first day Rade was indeed staying in an odd-numbered room. Then on the next day—the twelfth day—he must be in an even-numbered room. Thus, the impostor scholar Sula can go back and search again, starting from room 2 and in the worst case, by day 22, the thief Rade would be caught in room 12, and the stolen treasures retrieved.”
“Hmm…” After hearing Richard’s explanation, Great Scholar Socrates pondered for a long while and then looked at Richard, nodding, “Mhm, not bad, your line of reasoning is correct and pretty much identical to mine. You… ah, just one moment, let me draft a reply to that old scoundrel Adod.”
Having said that, Great Scholar Socrates picked up a quill, opened a new papyrus scroll, and began to “swoosh, swoosh, swoosh,” writing quickly.
After a while, having written most of his draft, Socrates reviewed the content and then fell into contemplation. Addressing Richard, he said, “Adod is deliberately making trouble with these tough questions for me, and although… ahem, although they haven’t truly troubled me, I should come up with a challenge of about the same difficulty in response to him.”
“I have thought of several challenges, but none of them seem quite right. Do you have any suitable question, preferably one that’s very difficult to solve…”
“Uh…” Richard’s eyes twinkled as his thoughts raced.
A very difficult problem to solve? There were plenty, and the one he had always wanted to know was one of them—what is the true nature of this world, what is the essence of transmigration?
Besides, some of the questions from long ago that tested the Book Spirit of the Monroe Chapter and caused it to remain unresponsive to this day were also applicable—like the Grand Unified Theory, the Riemann Hypothesis, and the exact value of pi.
However, considering these issues, since he, too, could not produce answers, it might be better to choose simpler ones instead. For example, the Poincaré Conjecture, which belonged to one of the seven great mathematical problems of the modern Earth world but had already been successfully resolved:
Any simply connected, closed three-dimensional manifold is homeomorphic to the three-dimensional sphere.
To put it simply, it means that every closed three-dimensional object without any holes is topologically equivalent to the surface of a three-dimensional sphere.
Or put even more simply, if you have an elastic band wrapped around an apple’s (or another spherical fruit’s) surface, by stretching it without breaking it or letting it leave the surface, you can move it gradually and shrink it to a point. However, if you bind the elastic band to a tire’s surface in a certain way, without pulling, it is not possible to shrink the band to a point without it leaving the surface. Therefore, the apple’s surface is “simply connected,” but the tire surface is not.
Richard was about to speak out but then paused, his words stopping at his lips, as he suddenly thought that the topic of topology might be a bit too challenging for the Great Scholar in front of him. If he truly mentioned it, he might first need to explain the definitions of dimensions, manifolds, and homotopy.
So… another simpler one then, preferably a pure numerical problem—lacking in any real technical content, yet requiring a significant amount of calculation to complete, a “laborious tough question.”
Then…
“One could think of it this way,” Richard said, addressing Socrates again. “There are peculiar existences amongst numbers, such as 121, 363, etc., called palindromes. These read the same forwards and backwards. And these numbers aren’t without basis; they can be decomposed into many other numbers.”
“For example, take the number 56. If you add it to its reverse, 65, you get the palindrome 121.”
“Another example, with the number 57, if you add it to its reverse, 75, you get 132. 132 is not a palindrome, but if you add it to its reverse, 231, you again get the palindrome 363.”
“Another one, add 59 to its reverse, 95, to get 154. Add 154 to 451 to get 605. Add 605 to 506 to get 1111—another palindrome obtained after three iterations.”
“Actually, about ninety percent of the numbers within 100 can yield a palindrome within seven iterations, with approximately eighty percent within four iterations.”
“Of course, some require more iterations, such as the number 89 which takes 24 iterations to reach the 13-digit palindrome, 8,813,200,023,188.”
“For numbers over 100, such as 10,911, it takes 55 iterations to reach a 28-digit palindrome—4,668,731,596,684,224,866,951,378,664.”
“And for an enormous number like 1,186,060,307,891,929,990, it took 261 iterations to form an acceptable palindrome, resulting in a figure exceeding 100 digits at 119 digits.”
“So is there such a number that, no matter how many times you iterate, you can never form a palindrome? We can call it a Lickerel Number. If it does exist, what is the smallest one?”
“…” Great Scholar Socrates fell silent, a long silence, and then he glanced at Richard. Silently, he moved to the side of the desk, picked up a cup of tea that had long gone cold, of which no one knew when it had been brewed, and took a sip.
After finishing the tea, Great Scholar Socrates looked at Richard, first nodding in agreement, “Mhm, quite a good question.”
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Then he asked two earnest questions.
