The Look-and-Say Sequence

The look-and-say sequence starts with 1, and each subsequent term is produced by describing the previous term out loud, run by run: "one 1" describes the term 1, so the next term is 11. "Two 1s" describes 11, so the next term is 21. "One 2, one 1" describes 21, giving 1211. "One 1, one 2, two 1s" describes 1211, giving 111221. The rule is always the same: scan the current term left to right, group consecutive identical digits, and write down the count followed by the digit for each group.

The puzzle value is in getting the rule right under time pressure — it is easy to misgroup digits or describe the count in the wrong order — and in noticing that the sequence's length grows geometrically: each term is roughly 1.303 times longer than the previous one (a constant discovered by John Conway, sometimes called Conway's constant), so despite looking like simple digit manipulation, the sequence explodes in length exponentially rather than staying compact.