Hailstone numbers
Pick a number. If it is even, halve it. If it is odd, triple it and add 1. Repeat. The numbers bounce up and down like hailstones in a cloud, and so far every number ever tried has eventually crashed down to 1. Try 27.
What's going on
This is the Collatz problem, and it is unsolved. Every number up to beyond 2⁶⁸ has been checked by computer and falls to 1, but nobody has proved that every number must. The great mathematician Paul Erdős said mathematics is not yet ready for such problems.
The rule only cares whether a number is odd or even, which is clock arithmetic with a 2-hour clock. Even steps shrink the number by half; odd steps roughly triple it. Since an odd step is always followed by an even one (3n + 1 is even when n is odd), you might guess the numbers tend to shrink. On average they do, but a single start like 27 still climbs to 9232 before falling.
Sheets: Clock arithmetic and Odd, even and things that never change.