The Simplest Math Problem No One Can Solve - Collatz Conjecture

- This is the most dangerous problem in mathematics, one that young mathematicians are warned not to waste their time on. It's a simple conjecture that not even the world's best mathematicians have been able to solve. Paul Erdos, a famous mathematician, said, "Mathematics is not yet ripe enough for such questions."

Here's how it works. Pick a number, any number. Seven? Good choice. Okay, we're gonna apply two rules. If the number is odd, we multiply by three and add one. So three times seven is 21, plus one is 22. If the number is even, we divide by two. So 22 divided by two is 11. Now, we keep applying these two rules. 11 is odd, so we multiply by 3, 33, and add 1, 34. Even, divide by two, 17, odd. Multiply by 3, 51, add 1, 52, even. Divide by two, 26, still even. Divide by two, 13, odd.

So we multiply by 3, 39, add one, and that's 40, which is even, so we divide by two, 20, divide by two, 10, divide by two, five, odd. Multiply by three, 15, add one, 16, divide by two that's eight, and then four, two, and one. Now, one is odd, so we multiply by three and add one, which equals four. But four goes to two, goes to one, so we're in a loop, and the lowest number is one. Now, the conjecture is this: every positive integer, if you apply these rules,

will eventually end up in the four, two, one loop. This is commonly called the Collatz conjecture after German mathematician, Luther Collatz, who may have come up with it in the 1930s. But the problem has many origin stories and many names. It's also known as the Ulam conjecture, Kakutani's problem, Thwaites conjecture, Hasse's algorithm, the Syracuse problem, and simply 3N+1. Why is 3x+1 so famous? - Among professional mathematicians, maybe it's not famous but infamous,

in the sense that if someone admits in public that they're working on it, then there's something wrong with them. (laughs) - [Narrator] The numbers you get by applying 3x+1 are called hailstone numbers, because they go up and down like hailstones in a thundercloud, but eventually, they all fall down to one, or at least we think they do. You can think of the numbers as representing the height above the ground in meters. So a number like 26 would start 26 meters above the ground.

And if you apply 3x+1, it rises up as high as 40 meters. And, in total, it takes 10 steps to get to one. So 10 is called its total stopping time. But take the very next number, 27, and it bounces around all over the place. In fact, it climbs all the way up to 9,232.

As an altitude, that is higher than Mount Everest, before it too falls back to the ground. In total, it takes 111 steps for 27 to get down to one and end up in the four, two, one loop. The paths that different numbers take vary so widely, even numbers right next to each other. So how do you even start to make progress on this problem? Well, honestly, mathematicians struggled. - People just decided that this was something invented by the Soviets to slow down U.S. science,

and it was doing a good job at it 'cause everybody's sitting there twiddling their thumbs and making no progress on this trivial thing that you can tell a school of children. - [Narrator] Jeffrey Lagarias is the world authority on 3x+1. - The first time I met him I was a senior in college, and he pulled me aside and he said, "Don't do this. Don't work on this problem. If you want to have a career, do not start spending time writing about this or publishing any papers about this.

Do real math for a while to establish yourself." - [Narrator] Alex Kontorovich didn't listen. He and Yakov Sinai looked at the paths of the hailstone numbers. Were there any patterns? But obviously all of them ended up at one. But what about the paths they take to get there? The pattern is randomness. Here is the sequence of a large number chosen at random. The graph peaks and then drop so low that you can't really see what's happening at this scale. But if you take the logarithm,

you find this wiggly graph with a downward trend. It looks like the stock market on a bad day. And this is no coincidence. Both are examples of geometric Brownian motion. That means if you take the log and remove the linear trend, the fluctuations are random. It's like flipping a coin each step. If the coin is heads, the line goes up, tails, it goes down. 3x+1 is just like the random wiggles of the stock market. Over long-enough periods, the stock market tends to trend upwards,

while 3x+1 trends down. Another way to analyze 3x+1 is by looking at the leading digit of each number in a sequence. Here are the hailstone numbers starting with three as the seed. And we can count up how many numbers start with a one, how many start with a two, how many start with a three, and so on to make a histogram. We can do the same thing for the sequence that starts with four, and that's a short one, and for the sequences that start with five, six, and seven. Again, for each sequence,

we're just counting up how many numbers start with each digit, one through nine, and adding that to our histogram. If you keep doing this for more and more numbers, eventually the histogram settles into a stable pattern. For the first billion sequences, you'll find one is by far the most common leading digit. 30% of all numbers start with one, around 17.5% start with two, 12% start with three, and the frequency decreases for higher digits. Fewer than 5% of all the numbers start with nine.

Now, this pattern is not unique to 3x+1. It comes up everywhere, from the populations of countries, to the value of companies, all the physical constants and the Fibonacci numbers, just to name a few. The distribution is known as Benford's law, and it is even used to detect fraud. If all the numbers on your income tax forms obey Benford's law, then, well, you're probably being honest. If not, you may be hiding something. In elections, Benford's law can be used to spot irregularities,