What is the Collatz conjecture?

0:47 Welcome to Matters Not Boring by kidopoly.com. I'm Mira and I'm Finn. Before we start exploring today, a very special happy birthday to Annie in Rio de Janeiro, who is celebrating today with her sister Jane. Annie loves exploring how things were made, and we hope you have the most wonderful birthday ever.

1:09 Happy birthday, Annie! And we also want to send a massive shout out to River in Los Gatos, California. River practices arithmetic on the daily drive and actually asked us all about today's idea.

1:23 If you haven't already, tap follow wherever you're listening so you never miss an episode.

1:28 So, River loves testing out maths puzzles in the car. Imagine driving on a strange, magical road. You can take any starting point you want. You can twist and turn as much as you like, but no matter what you do, you always end up trapped in the exact same endless roundabout.

1:48 And you cannot escape. You just keep driving in circles forever. It feels like a magic trick, but it is actually one of the biggest unsolved mysteries in the world. Wait, what do you mean trapped?

2:00 How can numbers get trapped?

2:02 To answer that, we have to travel back to 1937. A brilliant German mathematician named Lothar Kolatz loved looking for hidden patterns. He loved seeing what happens when numbers feed into each other, like turning on a machine.

2:19 Oh, like building a machine to see what comes out the other end. That is exactly the kind of stuff Annie loves figuring out on her birthday.

2:26 Exactly! Lothar Kolatz made a mathematical machine for positive whole numbers with just two simple rules. Rule one, if your number is even, you divide it by two.

2:39 Just chop it exactly in half. That is easy. If I have ten, I chop it in half and it becomes five. Perfect!

2:47 And here is rule two. If your number is odd, you multiply it by three and add one.

2:53 Okay, let me think. So my five is odd, three times five is fifteen, plus one is sixteen.

3:01 You got it! And then you just keep repeating those two rules on whatever new number you get. This beautiful, deceptively simple idea became known as the Colatz conjecture.

3:13 A conjecture is like a giant mathematical guess, right? Because nobody has proven it yet.

3:19 Yes, because what Lothar Colatz noticed happening with those two simple rules was so wild. Mathematicians are still scratching their heads about it. Nearly 90 years later.

3:31 Okay, I want to try it. Let's put a number in the machine and see what it does. Alright, let's track the number seven.

3:38 It is odd, so we multiply by three and add one. That shoots it straight up to 22.

3:45 Which is even. So we chop it in half and it drops to eleven. And from eleven, it takes a wild path.

3:52 It bounces up to 34, drops to 17, shoots up to 52, falls to 26, drops to 13, and shoots back up to 40.

4:04 Wait, it's just bouncing wildly. Up, down, up, way up, down. It is like watching a rubber ball bouncing down a rocky mountain. It is.

4:16 In fact, mathematicians actually call these hailstone numbers. Because inside a real storm cloud, hailstones get blown wildly up by the wind and drop down and get blown back up again before they finally fall to the earth.

4:31 Hailstone numbers. That is a brilliant name. But do they always fall down eventually? That is the incredible part.

4:39 If you try the number 27, it doesn't just bounce a little, it goes completely wild. It takes 111 steps of bouncing up and down. At one point, it blasts all the way up to 9,232 before gravity finally pulls it back down.

5:00 Whoa, a tiny number like 27 climbs to over 9,000. Just by multiplying by 3 and adding 1. Yes.

5:09 But no matter how wild the storm gets, no matter how high the number climbs, it always seems to shrink back down until it hits the number 4. And watch what happens there. From 4, it drops down to 2, and then falls to 1.

5:24 And from 1, the storm catches it again, blowing it straight back up to 4. Exactly!

5:31 And 4 falls to 2, 2 falls to 1, and you are right back at 4.

5:37 4, 2, 1. 4, 2, 1. You are trapped. That's the endless roundabout you were talking about.

5:47 That is the beautiful, strange trap of the Collats conjecture. Everything drains into it like a giant whirlpool. And the wildest part? It seems to happen to every single positive whole number in the universe.

6:01 Hold on, that cannot be right. What if you start with a massively huge number? Like a trillion? Surely one of them escapes the whirlpool.

6:11 You are thinking exactly like the world's best mathematicians. They didn't believe it either. They program supercomputers to check massive numbers, day and night, running the machine over and over.

6:25 And did any of them escape the 421 loop?

6:29 Not a single one. Every single number they threw at the storm cloud eventually came crashing down into the exact same trap. How many numbers did the computers check? They have checked over 295 quintillion numbers. That is 295 billion billions. Almost impossible to even picture.

6:55 295 quintillion? That is like counting every grain of sand on a giant beach. And every single one fell into the 421 trap.

7:06 Every single one. But here is the ultimate mystery, Finn. Because numbers go on forever to infinity, nobody has ever been able to prove that it will work for every single one.

7:19 So there might be a number out there, waiting in the deep darkness of infinity, that escapes the loop. Or maybe one that just keeps getting bigger and bigger forever and never comes down.

7:30 Exactly. A very famous mathematician named Paul Erdush once said about this puzzle that mathematics is not yet ready for such problems. It looks like simple arithmetic you can do in the car, but it hides a deep secret of the universe.

7:47 I am completely obsessed with this. I'm going to try it with random numbers today. I want to be the one who finds the number that escapes the hail storm. Alright, Finn, time for the quiz.

7:57 Are you ready?

7:58 Oh, I am so ready. Bring on the storm.

8:02 Okay, first question. In what year did Lothar Kollats come up with his famous mathematical machine? Next one. If you put an even number into the machine, what is the rule? Question three. Because the numbers bounce wildly up and down like they are in a storm cloud, what are they called? Alright, here's number four.

8:34 What are the three numbers in the endless loop that every tested number gets trapped in? Last question. How many numbers have supercomputers checked to see if they fall into the trap? Great job! Let's go through the answers and see how you did. The answer to question one is he came up with the idea in 1937.

9:02 For question two, the rule for an even number is to divide it by two or chop it in half. Number three, the answer is they are called hailstone numbers. Question four. The famous loop is four, two, one. And the final answer is supercomputers have checked over 295 quintillion numbers.

9:28 I still cannot get over that. Quintillions of numbers tested, and they all do the exact same thing. But nobody can explain why it has to be true forever.

9:38 It just goes to show how much magic is hiding right underneath the simplest edition and division. It is so beautiful.

9:46 Hey, if you enjoyed getting caught in the hailstorm with us today, please leave us a five-star review. Just scroll down and tap the stars.

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10:07 And if you want a shout out on the show like River or Annie, or just want to say hi, send us an email at hello at kidopoly.com.

10:15 See you next time on Maths Is Not Boring. Bye!