## I said simple, not easy!

The *Collatz Conjecture*, also known as the *3n+1* problem, has been driving mathematicians mad since the mid 1930s when it was it was first proposed by Lothar Collatz. Although it is simple to understand — as it involves nothing more than addition and division — it is not simple to prove.

Here’s the problem:

Take a given integer n. If it is even, divide it by 2. If it is odd, multiply it by 3 and add 1. Continue the pattern taking the answer as your new n.

Let’s look at some examples:

3, 10, 5, 16, 8, 4, 2, 1…

7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1…

12, 6, 3, 10, 5, 16, 8, 4, 2, 1…

21, 64, 32, 16, 8, 4, 2, 1…

You will notice that for each of these numbers, you end up at 1, which would lead to an infinite loop of 1, 4, 2, 1, 4, 2, 1…

*SEE ALSO: Mathematician Solved a Nearly Impossible Maths Problem*

But is that the case for all integers? That’s the Collatz conjecture, and it has never been proven. It has been tested for absurdly large numbers — for example, the largest number of steps it takes for a number below a billion to reach 1 is 986 — but nobody has been able to say for sure that no number exists that won’t ever reach 1.

H. S. M. Coxeter wrote of the difficulty of finding a counter-proof to the conjecture: “I must warn you not to try this in your heads or on the back of an old envelope, because the result has been tested with an electronic computer for all x_{1} ≤ 500,000. This means that, if the conjecture is false, the prizewinner muse [sic] either find a sequence of this kind which he can prove to be divergent, or else find a cyclic sequence for this kind whose terms are all greater than half a million.”

But, if you are still insistent, there is some prize money involved. Paul Erdos said of the conjecture: "Mathematics is not yet ready for such problems," but he offered $500 for a solution. Thwaites, after whom the problem is sometimes called the Thwaites’ conjecture, offered up £1000 (about $1500) as well.

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