1, 11, 21, 1211, 111221... what comes next in this sequence?
Start with 1. To build the next line, read the previous line aloud and write down what you say: '1' becomes 'one 1', so line two is 11. That becomes 'two 1s', so line three is 21. Keep going: 21, 1211, 111221... What's the rule for generating any line, and does the sequence ever settle into a repeating pattern?
Reveal the answer
Each line just describes the digit-runs of the one before it ('one 2, one 1' -> 1211, and so on), so it never repeats — it grows forever. John Conway proved in 1986 that every such sequence eventually breaks down into a fixed set of 92 recurring 'elements' (which he named after chemical elements) and that its length grows by a constant factor each step, now called Conway's constant (≈ 1.303577).