Collatz Journal - A Math Expedition

I'm using the #100Days challenge to blog my way toward a book for math amateurs exploring the Collatz Conjecture.

I'm calling it: Collatz For Crackpots

That's a title that requires two definitions:

What is Collatz? and Who/What are the Crackpots?

First, the Collatz Conjecture has been called the "Easiest, Hardest Math Problem". You can explain it to any precocious 8-year old as follows:

• Take any number N.
• If it is even, divide it by 2.
• If it is odd, multiply by 3 and add 1.
• Repeat.

What happens? For example, try 3:

3 is odd so it goes to 3*3+1 = 10
10 is even so it goes to 10/2 = 5
.. and so on.

Here's how it plays out:

3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1

Easy enough. It goes up and down but winds up at 1.

Now try 7:

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

Interesting. It also goes up and down but ends up at 1.

Try a few numbers and you'll see similar short patterns that bounce up and down but seem inevitably to arrive at 1. Now, try 27:

27 -> 82 -> 41 -> 124 -> 62 -> 31 ... and keeps going

Ultimately, it takes 111 steps and rises to 9,232 (!) before falling back 1.

So, the Collatz Conjecture asks:

Do ALL numbers run through the Collatz sequence eventually end up at 1? Or, is it possible there's a number (or numbers) that rise forever? Is there a number (or numbers) that rise and fall in an infinite loop?

Here's where "Crackpots" come in. This problem is so approachable by anyone with basic math knowledge that many are tempted to tackle the Conjecture. It attracts enthusiasts and tinkerers who think they've got a new approach but here's the trap:

This problem was introduced by German mathematician Lothar Collatz in 1937. Since then it has been tackled by hundreds, if not thousands, of professional mathematicians, millions of amateurs. Powerful computers have run the sequence on every number up to 2⁶⁸ (that's 1 followed by 21 zeros in decimal) and they've ALL (billions and billions of numbers) returned to 1.

Still, billions and billions of numbers are evidence and not proof that there isn't a 35 digit number (or millions of as-yet-unfound numbers) that could loop or rise infinitely.

The approachability of this problem (plus the existence of a $700K prize) means that the Internet is awash in forums with "enthusiasts" who claim to have solved the Conjecture. Yet, many of these explorers are neither equipped with a common set of math tools to solve the problem nor a solid idea of how to express a solution.

With this series of blog posts, I hope, as a proudly-proclaimed "math misfit" myself, to provide an well-illustrated map to the Collatz territory and give my fellow crackpot explorers robust tools and concepts to use on their journey.