Examine carefully the following sequences of numbers: 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221, 13211311123113112211. Although the sequences appear to behave totally at random, in fact, after the first sequence, each sequence is constructed in a precise and methodical way based on the previous one. What is the next sequence?

Bryce

Starting with the second sequence, each sequence is simply a detail description of the previous one. So, for example, the second sequence describes the first: one one, that is, 1, 1. The third sequence describes the the second: two {ones;} or 2, 1. The fourth describes the third: one two, one one: or 1, 2, 1, 1. This sequence in turn is described as follows: one one, one two, two ones (111221). The sequence that follows is therefore 312211 since it describes the exact composition of the preceding sequence. All of the above means that, in the series of sequences given, the next sequence is 11131221133112132113212221. Nov 03 02, 4:31 AM