Learn about why math is a mystery just waiting to be solved
The different routes each number takes within the Collatz Sequence
The pen falls with a clink, hitting the shiny wood abruptly. My eyes roll in annoyance, the second hand on the clock seems to move much slower than usual. Until, finally, it clicks. Everything makes sense. The years of frustration, the hours of pure agony, everything finally clicks. The mystery has been solved. The culprit is never the one you originally suspected.
I’m pretty sure that’s what it would feel like to finally solve the Collatz Conjecture.
Unfortunately, mathematics hasn’t gotten there yet.
Now, you may be wondering: What is Nina talking about? What is the Collatz Conjecture, and why did she sound like a Sherlock Holmes book?
All will be answered in good time; I promise.
First, let me explain. To understand the answers to these questions, open up your brain canvas. Let’s start with a function that states that for any inputted positive integer, if even, it should be divided by two, and if odd, it should be multiplied by three, and one should be added. For example, if the input is three, the output would be ten, and if the input is eight, the output would be four.
The conjecture claims that if you keep repeating this process for any given number’s outputs, you will always come to a value of one. For example, if you were to input three, you would get ten, then five, then 16, then eight, four, two, and one. And when you get to one, you can easily get back: one becomes four, which becomes two, which becomes one again (1).
There’s a lot more theory here than it seems: there are parts of this sequence called ‘cycles’ or loops from a starting number back to itself. Of course, in that case, unless the cycle included one, the Collatz Conjecture would be false, as the numbers in the cycle would not get to one given that they would only loop around. The only confirmed cycle we know to be in existence is from one to four. One is made into four which is made into two which is made right back into one.
The Collatz Conjecture is deceptively elusive, though. While there is glaring evidence that it works, from logic-based proofs to brute-force solutions for every number, no one has officially found a way to prove this for all integers yet. It’s the same thing as having circumstantial evidence to indict your crime scene suspect but not enough to definitively prove them guilty. And therefore, in math-speak, it is untrue (for now). Of course, as with many problems, there are some proofs that have come close.
For example, we have realized that all even numbers will eventually turn into an odd number, given they will keep dividing by two, so it is only needed to check whether all odd numbers will enter the 4-2-1 cycle eventually, and therefore have a final result of one (2). Computer simulations have been able to prove this for many numbers, the largest being 2100000 – 1 (3).
This is math! It is made up of mystery and riddles and passion. It is made up of unsolved problems that you too could solve if you put in hours and your original thoughts. The Collatz Conjecture is just one of many possibilities.
- Collatz problem. Wolfram MathWorld. (n.d.). Retrieved from https://mathworld.wolfram.com/CollatzProblem.html
- Furuta, M. (2022). Proof of Collatz Conjecture Using Division Sequence. In Advances in Pure Mathematics (Vol. 12, Issue 02, pp. 96–108). Scientific Research Publishing, Inc. Retrieved from https://doi.org/10.4236/apm.2022.122009.
- Collatz conjecture for 2^100000-1 is true – algorithms for verifying extremely large numbers. IEEE Xplore. (2018). Retrieved from https://ieeexplore.ieee.org/document/8560077.
- Hartnett, K. (2021, September 10). Mathematician proves huge result on ‘dangerous’ problem. Quanta Magazine. Retrieved from https://www.quantamagazine.org/mathematician-proves-huge-result-on-dangerous-problem-20191211/
The sequence mapped out for numbers 1 to 100,000! Not very granular but cool to see how each number progresses.
A cool-looking stranded image that represents routes of numbers within the Collatz Sequence. I’d recommend not reading too much into the placement of different strands though :).