Here’s a classic brainteaser that I don’t like. What’s the next number in this sequence: 1, 11, 21, 1211, 111221, …? The answer is 312211, because each number describes the digits in the number that precedes it. We open with 1, an arbitrary choice, but the next number describes 1 as “a single one,” i.e. “one one,” i.e. 11. The next entry describes 11 as “two ones,” or 21. This, in turn, is “one two followed by one one,” or 1211, and so on.
The first digit indicates that the number contains two 0s, the next indicates zero 1s, the next indicates two 2s, and the final indicates zero 3s. Bonus: you can seed the look-and-say sequence with any whole number. For example, if you started with 39, then the next entry would be 1319 . Conway proved that all seeds yield a sequence whose entries grow to infinity, with only one exception. Find the exception. I’ll be back next Monday with the solutions and a new puzzle.
John Horton Conway Sam John Conway Sequence Elementary Arithmetic Look-And-Say Sequence Subtraction Addition Gizmodo
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