The Collatz Conjecture
Pick any number. Even, halve it. Odd, triple it and add one. It always falls to 1, and nobody can prove it.
Your birth year. Your age. Your phone number. Anything. I will run the rule and trace the fall.
Tip: hover a number to see why it is famous. The smallest ones often fall the hardest.
The reframe
You just watched a number you chose survive a fall that nobody can guarantee. That is the strange heart of this toy. There is overwhelming evidence the conjecture is true: every number anyone has ever fed in, billions of billions of them, all the way past 2 to the 68th power, has come home to 1. And there is, after almost ninety years, not one shred of proof that it must.
That gap, between true and proven, is the thing I want you to feel. We live most of our lives in it. You are almost certainly safe on the next drive; nobody can prove it. The bridge will almost certainly hold; the proof is statistics, not certainty. Mathematics is the one place that promised to close the gap completely, and here is a rule a ten-year-old can follow that the whole field still cannot.
So when your number clawed upward and you wondered, just for a second, whether this would be the one that escapes to infinity and breaks the pattern forever, that flicker of doubt was the honest response. Not knowing is not the same as it not being true. It just means we are not finished looking.
The history
The German mathematician Lothar Collatz wrote the problem down in 1937, and it has collected names ever since: the 3n+1 problem, the Ulam conjecture, the hailstone problem, the Syracuse problem. Paul Erdos, one of the most prolific mathematicians who ever lived, said mathematics is not yet ready for such problems and offered 500 dollars for a proof; the money has never been claimed. In 2019 Terence Tao proved that almost all starting numbers eventually drop below their starting point, the strongest result so far, and a careful reminder that almost all is not all. Computers have now verified every number up to roughly 2 to the 68th power. Each one falls. The conjecture remains open. It is, in the words of more than one mathematician, the simplest impossible problem in the world.