The Collatz Conjecture

Right. Imagine this.

You’ve had a long day. You’re not quite ready for bed. You’ve got that dangerous mix of curiosity and poor decision-making.

So you invent a maths game.

Not a proper, sensible one… but one of those “this feels too simple to be interesting” ones.

Welcome to the Collatz Conjecture, possibly the most chaotic “simple” idea ever created.

The Rules (Deceptively Harmless)

Pick any positive number. Any number. Go on.

Now follow this:

If it’s even → divide it by 2
If it’s odd → multiply it by 3 and add 1

Repeat.

That’s it.

That’s the whole thing.

No tricks. No hidden clauses. No “terms and conditions apply”.

Let’s Have a Go

Start with 6:

6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1

Done. Easy.

Start with 11:

11 → 34 → 17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1

Still ends at 1.

You’re probably thinking:

“Right, cool… everything just sort of settles down eventually.”

Yeah.

Everyone thought that.

The Problem (AKA Where It Gets Weird)

The conjecture says:

No matter what number you start with, you will ALWAYS end up at 1.

Always.

Every time.

Forever.

The Catch

No one has ever proven it.

Let that sink in.

We’ve:

Built nuclear reactors
Landed on the Moon
Invented TikTok (arguably a step backwards)

…and yet…

This little “divide by 2, multiply by 3 and add 1” thing?

Still unsolved.

It Gets Worse

Some numbers behave… strangely.

Take 27.

You’d expect it to calmly head down to 1.

Nope.

It goes on an absolute bender:

27 → 82 → 41 → 124 → 62 → 31 → 94 → 47 → 142 → 71 → 214 → 107 → 322 → 161 → 484 → 242 → 121 → 364 → 182 → 91 → 274 → 137 → 412 → 206 → 103 → 310 → 155 → 466 → 233 → 700 → 350 → 175 → 526 → 263 → 790 → 395 → 1186 → 593 → 1780 → 890 → 445 → 1336 → 668 → 334 → 167 → 502 → 251 → 754 → 377 → 1132 → 566 → 283 → 850 → 425 → 1276 → 638 → 319 → 958 → 479 → 1438 → 719 → 2158 → 1079 → 3238 → 1619 → 4858 → 2429 → 7288 → 3644 → 1822 → 911 → 2734 → 1367 → 4102 → 2051 → 6154 → 3077 → 9232 → 4616 → 2308 → 1154 → 577 → 1732 → 866 → 433 → 1300 → 650 → 325 → 976 → 488 → 244 → 122 → 61 → 184 → 92 → 46 → 23 → 70 → 35 → 106 → 53 → 160 → 80 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1

It shoots up to over 9,000 before finally giving up and crawling back down.

Absolute chaos.

Why Does This Matter?

Honestly?

No one really knows.

It doesn’t obviously help with:

Engineering
Finance
Climate modelling
Or even deciding what takeaway to have on a Friday

But mathematicians are obsessed with it.

Because it should be easy.

And it absolutely isn’t.

It’s like that one Parkrun where you think:

“I’ll just take it steady…”

…and suddenly you’re redlining in kilometre 2 wondering where it all went wrong.

The Haphazard Take

The Collatz Conjecture is basically the mathematical version of:

“Just one more episode”
“Quick 5k run”
“I’ll just check TikTok for a minute”

Simple start.

Completely unpredictable middle.

Eventually ends somewhere familiar… but not before things get weird.

Final Thought

Somewhere out there is a proof.

A neat, tidy explanation that makes all of this make sense.

But until then…

We’ve got this beautifully chaotic little rule set that:

anyone can understand
no one can solve
and everyone underestimates

Take the risk. Multiply by 3 and add 1.
Make the mess. Divide by 2 when it all gets too much.

But whatever you do…

Stay Haphazard.

FOOTNOTE
I’m no mathematician. Not even close. I mean, I still count in my head when I’m out on a run trying to pace a 5K… and half the time I get that wrong. Hell, I’m barely even an athlete more of a participant with ambition and questionable decision-making.

But that’s kind of the beauty of something like the Collatz Conjecture. You don’t need a PhD, a lab coat, or even a functioning attention span to get stuck into it. It’s simple enough to try on the back of a fag packet, yet somehow deep enough to keep actual geniuses awake at night. A bit like signing up for a “casual” Parkrun and ending up in a full-blown personal crisis by kilometre three.

If you fancy going down the rabbit hole (and fair warning, it does spiral), there are loads out there that explain it far better than I ever could. Videos, articles, proper mathematicians doing proper thinking… all the good stuff. This is one of those topics where five minutes of curiosity can turn into an hour of “wait… what just happened?”

So if you want to learn more about this snizzle, definitely go have a look around. Just don’t blame me when you find yourself scribbling numbers at midnight, wondering why 27 behaved like it had a few too many at the pub and gone home a shat the bed.