They tell us that mathematics is useful, that it is important because with them we can understand from the universe how a medicine affects us, but that is not the whole truth. Others tell us that mathematics is marvelous for its beauty, for how they entangle themselves in complicated reasoning, but they are not telling everything. There is another reason to love mathematics, one of the reasons that has convinced more people to study them: they are fun.
They are, this is so. Recreational mathematics exist and are riddles, challenges that put our brain to work. You just need pencil and paper, or sometimes not even that. Moreover, there are times when this spirit of recreational mathematics permeates the rest of the areas and that is when they become viral among professionals. So, let’s play something.
Let’s play something
Take a positive integer (those we use to count, without decimals or frills) Pay attention, because we are going to define an algorithm, which, in other words, is just a way of making decisions. If your number is even, divide it by two. If instead, your number is odd multiply it by three and add one. Now you have a new number, right? Then apply the above algorithm again, if it is even, divide it by two and if it is odd, multiply it by three and add one. Take the result and process it again, as if you were kneading the number, over and over again. If you wonder how many times you have to do this, the answer is simple: you continue, you will notice when to stop.
If you have continued playing with your number with almost all probability you will have entered a loop. At some point your algorithm will have returned the 4, which being even you will have divided by two getting a 2, which as it is even you will have divided by two getting a 1, which is odd, so you should have multiplied it by three and having added one, getting the 4 again. You are in an endless cycle, once you have fallen there is no way out of it, but could we have avoided it? Is there any way to escape from 4 2 1? That is exactly what the mathematician Lothar Collatz wondered in 1937. Since then more than 80 years have passed and we still do not know if all roads lead to 1.
Brute force is not the solution
As we commented in the article on the goldbach conjecture, mathematics tries to be as accurate as possible and they are not worth approximations when it comes to considering a statement true or false. If we accept that the Collatz conjecture is true and that all numbers end up falling into the 4 2 1 loop, then we better comply with all the infinite integers that exist and, unfortunately, trying them all is not a option.
What can be done is to find an exception. In order to discover that there is a number that does not enter that loop, it will be sufficient to prove that the Collatz conjecture is false. On the other hand, you can try to find some contradiction, something like that: if the conjecture is false (or true) some conclusions would be derived that would contradict more fundamental principles of mathematics, much better established foundations.
The problem is that it has been tried, and not by hand, but with computers much more powerful than our brains armed only with pencil and paper. There is no number that is free of the loop between approximately the first 10 ^ 18, or said in a traditional and worryingly long way the first 1 000 000 000 000 000 000 what in Castilian we call a trillion (not to be confused with the English trillion , which is our trillion, nor this one with the English trillion, which is our billion, but hey do we continue?)
The beauty of a good graphic
Although the brute force approach has not resulted (which is what this type of approximation is called where many numbers are checked one by one) has left us some very interesting graphs, such as this one, where the horizontal axis numbers are shown 1 to 10 000 and in the vertical how many times we have to repeat our algorithm until we fall into the loop. It is that messy order that arises in so many mathematical problems. A beautiful and suggestive distribution, but there are many more.
Others prefer to use trees where they follow the opposite path. Starting from 1, they retrace their steps simulating how all numbers are obtained from it, like the branches of a plant. Normally the branches seem distributed in any way, but there are those who look for increasingly creative ways to structure them, for example, by tilting the odd branches to one side and the right to the other, getting structures that, they say, that they resemble living organisms, such as algae or corals. Of course, it is pure pareidolia, but surprisingly suggestive. The longest branch we know is the number 75,128,138,247 that needs nothing more and nothing less than 1228 steps to reach 1.
But what would a number that Collatz did not meet?
The first time you hear about this conjecture, it is normal to ask yourself how you can be sure that a number never reaches 1. Does this mean that it keeps changing its values over and over again infinitely? And if so, how do we distinguish that a number needs millions of millions of interactions from another number that simply never comes? After all, there are unimaginably large finite numbers.
The answer is simple, it is a misunderstanding. If you think about it when we apply the Collatz algorithm to a number, if the conjecture is correct, it should not retrace its steps (unless it is 1, 2 or 4). If it did, it would be falling into a loop, so at the same time when using the Collatz algorithm you get a repeated number, congratulations, you will know that you have found the desired counterexample. For the Collatz conjecture there is only one known cycle, the famous 4 2 1, and in fact it is called the “trivial cycle”.
Meanwhile, the numbers will go up and down like crazy, going through all the digits they want. In fact, for that reason they are called “hail numbers,” because, hail is formed when a stream of air ascends raindrops to freeze, causing condensation to accumulate on the surface that makes it weigh more, throwing it back to earth. Sometimes, a new current rises to the hail back to the heights, repeating the scene a few times, but it always ends on the ground, at number 1.
Along with Goldbach’s conjecture this is another one of those mathematical problems that can sink a career. Seemingly simple questions, but so difficult that they can absorb a lifetime without getting the slightest advance. Their simplicity, the beauty of the statements and especially the legend that has been built around them, make these mathematical puzzles a temptation difficult to tame. One of the greatest mathematical geniuses of our generation, Terence Tao, recognizes that it is not allowed to work on these types of questions more than one or two days a year, so powerful are the songs of the mathematical sirens.
Because the numbers, although it may seem otherwise, hide hobbies capable of abstracting you as if time had vanished, swallowing your minutes, your hours and even your years. Only you and the role in a fight against history.
DON’T KEEP IT UP:
- No one has been able to prove Collatz’s conjecture, but neither has it been proven that it is impossible to achieve (that it is undecidable).
- What Terence Tao got was to demonstrate that, approximately (and that is the key word), all the numbers end in the trivial cycle.
- Lagarias, Jeffrey C. “The ultimate challenge: the 3x + 1 problem.” American Mathematical Society 2010
- Simons, John L. “On the nonexistence of 2-cycles for the 3x + 1 problem. ” Math Comp. 74: 1565–72. 2005
- Terence Tao “Almost all orbits of the Collatz map attain almost bounded values” arXiv. 2019