The sequence that describes itself

Question

Consider this sequence: 1,\ 11,\ 21,\ 1211,\ 111221,\ \dots The numbers grow, but not by any addition or multiplication rule you'll find by looking at differences or ratios. What is the next term? (Hint: try reading each term out loud.)

Accepted Answer

The next term is 312211. The rule: read the previous term aloud. Each term describes the digits of the term before it, by saying how many of each digit appear in a row ("look and say"): - 1 is read as "one 1" → \mathbf{11}. - 11 is "two 1s" → \mathbf{21}. - 21 is "one 2, one 1" → \mathbf{1211}. - 1211 is "one 1, one 2, two 1s" → \mathbf{111221}. - 111221 is "three 1s, two 2s, one 1" → 3\,1\ 2\,2\ 1\,1 = \mathbf{312211}. So reading 111221 out loud gives "three 1s (31), two 2s (22), one 1 (11)," which strings together to 312211. Why it stumps people. Everyone hunts for a numeric pattern, differ…

The sequence that describes itself

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