The world of mathematics is no stranger to unsolved problems; there are many loaded questions which we may never find an answer to. Every day we discover that despite all our knowledge, there are more questions for which we lack answers. Out of all of them, one has stood out for years, and it may appear surprisingly straightforward. Despite its apparent simplicity, it has managed to baffle mathematicians for decades, with many agreeing that it’s a problem that ‘beats’ our modern understanding of mathematics.
This is the infamous Collatz Conjecture, named after mathematician Luther Collatz who is believed to have first proposed it in the 1930s. He suggested an algorithm, with two possible processes: You start with a positive integer, and if it’s even, you divide it by two, so x/2. If it’s odd, then you multiply by three and add one, so 3x + 1. You repeat this process until you get to one; which Collatz argued was a certainty. This has remained unrefuted since the conjecture was first introduced in 1937 and yet, we still don’t know why or even if it’s valid.
Let’s try starting with the number 15 for example. 15 is an odd number, so applying 3x+1, we get 46. 46 is even, so we divide by two, and we get 23. Then 70, then 35, 106, 53… until we reach 1 after a total of 17 calculations. The numbers from 15 to 1 are all labeled as “hailstone numbers” and we call the number of calculations it took us, 17, its “stopping point”. However, if we start out with 16, it only takes 4 calculations to reach one. 17 takes 12, 18 takes 9. There is not a clear relation between the routes each integer takes before reaching 1.
Even though we haven’t found a consistent pattern, every positive integer up to 2⁶⁸ has been tried for the conjecture using computers; every single one has been proven to adhere to it. This still is not definitive however, as there may be counterexamples when dealing with extremely large numbers. Besides experimental results, proof about stopping times and integer ranges have been proposed. Still, after almost a century of working on this problem, we have not found an answer. This has not discouraged everyone, and there are still experts dedicating their lives to hopefully uncovering the mystery of this problem. Why?
They believe understanding the algorithm may help advance mathematics as a whole. The Collatz Conjecture is not exactly a standalone problem; just as many other mathematical problems aren’t. It’s deeply rooted in number theory and algorithms, and our current knowledge of maths has proven ineffective to find a solution. Thus, many mathematicians hope that with solving the conjecture, we can find the missing jigsaw piece, our knowledge gap in mathematics, and be one step closer to understanding the universe.
In conclusion, although we have yet to prove or disprove the conjecture, math remains hopeful. This seemingly simple algorithm may help us learn a lot more about mathematics and the world around us. Perhaps with understanding the Collatz Conjecture & finding the missing piece of the puzzle, we just might be able to see the full picture forming in front of us.
