The Great Mathematical Problems

Transcription

Pure mathematics, that is, math for its own sake, has produced fascinating patterns, such as erie strange attractors, or tables of knots. Applied mathematics has been used in many areas, such as heat flow or turbulence. There's one problem, however, which leaves mathematicians utterly defeated. And it only involves simple arithmetic that a seven-year-old can follow. This is definitely the simplest impossible problem. Starting with a positive whole number n, let's produce a new number according to the following rule: if n is even, divide it by 2. If it's odd, multiply by 3, then add 1. For example, let's start with 10. Since 10 is even, divide it by 2 to get 5. 5 is odd, so multiply it by 3 and add 1 to get 16. Keep going to produce 8, 4, 2, 1, 4, 2, 1, etc. This pattern of 4,2,1 repeats forever. Well, this isn't hard. So what's the problem? Try other starting numbers, like 11, 23, or 29. They all eventually reach one. This is the challenge; show that no matter which starting number you choose, the numbers will always reach one. This problem drives mathematicians crazy because there don't seem to be any clear patterns. Sure, some special numbers, such as 8192, which is a power of 2, collapse down to 1 pretty quickly; it takes only 13 steps to get there. However, if you start with 27, it takes 110 steps to reach the number 1. A graph of the points when we start at 27 shows the erratic nature of these numbers. The graph reaches its peak at 9232. Of course researchers have used computers to help out. You can click on this box to enter your own starting number and explore what happens. To date, all starting numbers less than 5x2^60 have been checked and they all eventually reach one. Of course this doesn't prove the conjecture for larger starting numbers, but it does mean that working by hand is not a good idea. This impossible problem is usually called the 3x+1 problem, but it's also known as the Collatz Conjecture, named after Lothar Collatz who invented the problem back in the 1930s. Other mathematicians who were intrigued by the problem mentioned it in their lectures, so this conjecture also became known as Hasse's problem, Kakutani’s Problem, and Ulam's problem. With all this interest, it was joked in 1960 that the 3x+1 problem was part of a conspiracy to slow down mathematical research in the U.S. But getting back to the problem, what could happen if a starting number doesn't reach the cycle {4,2,1}? One possiblity is that it approaches some other cycle. Advanced theory shows that any cycle besides {4,2,1} must have at least 10 billion numbers. The only other possibility is that the numbers would get arbitrarily large and approach infinity. But both of these scenarios are highly unlikely. Over time, mathematicians have built complex theories to try to understand the 3x+1 problem, but they've made little progress. Even the 20th century genius Paul Erdos said about this challenge, "Mathematics is not yet ready for such problems". But hey, but don't let me or Paul discourage you. What can you see in this problem?