The Collatz Conjecture – Deceptively Simple

Let’s do a magic trick.

Pick a number, any number. Now if it’s even, divide it by 2. Otherwise, if it’s odd, multiply it by 3 and add 1. Repeat this process over and over until you get to 1.

For example, if we start with the number 6, we would perform the following steps:

1. 6 is even, so divide by 2: 6 / 2 = 3
2. 3 is odd, so multiply by 3 and add 1: 3 * 3 + 1 = 10
3. 10 is even, so divide by 2: 10 / 2 = 5
4. 5 is odd, so multiply by 3 and add 1: 5 * 3 + 1 = 16
5. 16 is even, so divide by 2: 16 / 2 = 8
6. 8 is even, so divide by 2: 8 / 2 = 4
7. 4 is even, so divide by 2: 4 / 2 = 2
8. 2 is even, so divide by 2: 2 / 2 = 1

I can guarantee that no matter the number you pick, you’ll end up at the number 1 after a while.

You may not realise why this is surprising. However, if you think about it, there should be no good reason as to why we end up at 1: after all, multiplying a number by three and adding a bit has a bigger effect than dividing by two. We could expect that at least some number ends up blowing up to infinity, no?

Turns out, if we run this on a computer, we find that every number we test ends up at one. However, this doesn’t prove anything. Mathematicians search for formal proofs for any number, and are not happy with just using a trial-and-error method. After all, if you test a bazillion numbers, how do you know that the bazillion-and-first doesn’t break the rule?

This little mystery is what’s knows as the Collatz Conjecture, or the 3n + 1 conjecture. It is a mathematical problem that has puzzled mathematicians for over 80 years. The conjecture is named after German mathematician Lothar Collatz, who first posed the problem in 1937. It’s a simple problem with seemingly complex solutions, and it’s a great example of how math can be both fascinating and frustrating.

Despite its simplicity, no one has been able to prove or disprove the conjecture. In fact, it’s been tested on millions of numbers, and it always seems to hold true. But without a proof, mathematicians can’t be certain that it’s true for all numbers.

So, why is this problem so difficult? Well, it’s because it’s recursive – meaning that each step depends on the previous step. This makes it hard to analyze and predict what will happen next.

There have been a few attempts to prove the conjecture, but none have been successful. In the 1980s, a team of mathematicians tried to prove it using computers, but their calculations were so complex that they ended up crashing the computer.

Despite its simplicity, the Collatz Conjecture has many interesting properties and connections to other areas of mathematics. For example, it’s closely related to the mathematical field of dynamical systems, which studies the behavior of systems over time. It’s also connected to the study of prime numbers and the distribution of prime numbers in the natural numbers.

Through the years, the Collatz Conjecture continues to fascinate mathematicians and puzzle enthusiasts alike. It’s such a legendary problem that young mathematicians are repeatedly told not to waste their time on it, as it’s easy to be deceived by its apparent simplicity! After all, the conjecture has confounded some of the greatest minds in mathematics since its conception. Whether or not it will ever be solved remains to be seen, but it continues to captivate and challenge those who dare to take it on.