The Collatz conjecture
Pick any positive whole number and apply one simple rule over and over. The conjecture says you always end up at 1, no matter where you start. Here the height of each ball is the size of the number.
Two rules
If the number is even, halve it (blue). If it is odd, triple it and add 1 (orange). Odd steps jump up, even steps drop down.
Starting at 7
7 is odd, so it becomes 22. 22 is even, so it becomes 11, and 11 jumps to 34. The sequence zigzags unpredictably.
Hailstone numbers
Like a hailstone tossed up and down in a cloud, the values rise and fall. Starting from 7, the sequence climbs as high as 52 before it starts to come down.
Down to 1
Once the sequence lands on a power of two, like 16, it just halves all the way down: 16, 8, 4, 2, 1. Then 1 becomes 4, and the loop 4 → 2 → 1 repeats forever.
Paths merge
Different starting numbers join the same road. 6 goes through 3 to 10, and 9 goes through 28 and 14 to 7. All the paths seem to flow together like streams into one river ending at 1.
Still unproven
Computers have checked every starting number up to about 10²⁰, and all reach 1. But no one has proved that it holds for every number: some sequence could, in principle, grow forever or get stuck in another loop.