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What is the next three numbers for this pattern? 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221, 1321131112311

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2 Answers

You have stumbled upon a series that I used to give my students as a fun challenge!  If you really are stuck, I'll give the answer below, but don't look if you still want to give it a shot!
The first number is 1, so say "one."  Now, to form the next number, tell me what I have in the previous one---"one one"---so write 11.  Then, the next consists of "two ones", so write 21.  The next consists of "one two, one one", so write 1211, etc.  Keep reading the previous number in terms of blocks of any given digit, and you'll have your sequence.
Very smart-alecky type of recursive sequence, and it always elicits a groan from the audience once they hear the answer.
Hassan H.