The rule fits in just two lines. If a number is even, divide it by 2; if it's odd, triple it and add 1. Apply the same rule again to whatever new number comes out. Start from 3, for example, and you get 3→10→5→16→8→4→2→1, reaching 1 in 7 steps. This problem, first posed in 1937 by the German mathematician Lothar Collatz, is especially famous because the rule is simple enough for an elementary school student to understand.
The Collatz conjecture is the claim that "no matter which natural number you start with, keep repeating this rule and you'll eventually reach 1." Computers have checked every number smaller than 2⁶⁸ (about 3×10²⁰), and not a single exception has been found. But here's the crucial point — that only means it's "true within the range checked so far." No one has yet managed to prove that it's always true for every natural number. The Collatz conjecture remains an unsolved problem in mathematics to this day.
If the rule is this simple, why hasn't it been solved yet? It's because the rule works completely differently depending on whether a number is even or odd, and there's no good mathematical theory for handling a number growing and shrinking at the same time. Being a small number doesn't make it easy, either. 27 is only a two-digit number, yet it takes 111 steps to reach 1, climbing as high as 9,232 along the way. The rule is predictable, but the actual path it takes bounces around almost as if at random.
The famous Hungarian mathematician Paul Erdős went so far as to say, of this problem, that "mathematics simply isn't ready to handle a problem like this yet." It also carries a prize, so mathematicians and amateurs around the world are still taking it on today. Some mathematicians believe "an exception that falls into some loop other than 1 might eventually be found," while others believe "it will eventually be proven" — but neither side has a definite answer yet.
On our activity page, you can pick a starting number and follow the calculation process one step at a time with a slider, watching even numbers (blue) and odd numbers (coral) alternate with your own eyes. You can also compare, on a graph, the number of steps it takes to reach 1 and the highest value reached along the way for every number from 1 to 30, and find standout numbers like 27 for yourself.