13112221 - Number sequence - 1113213211

Have a go at the problem. Can you work out the next number?

1
11
21
1211
111221
312211
13112221
1113213211

Here is the start of the sequence, on the left. You start with "1". Then each line describes the line above. So the second line says that there is one "1" above. This is "11". Now the third line says that there are two '1's above, or "21". The fourth line says one "1", then one "2". Of course, all the numbers are digits rather than words, and you leave out 's' (so two "1"s becomes "21").

You might get annoyed at this, but it does teach you that you can look at numbers in different ways. Sometimes the digits are used to describe an amount and sometimes they are just a symbol (from the previous line).

There are interesting things about this sequence. You never get digits more than "3" in it (if you start from "1"). To get "4" as a digit, for example, it must describe a number of digits the same in the previous line (it can't just BE a symbol in the previous line, as then the '4' would go all the way back to the first line, and it isn't there). So they must be 4 digits together on some line. Well, there are two ways that you could do this. Say you had "42". This means that the previous line had "2222". This could be 2x2, 2x2. But that is wrong, as it should be described as 4x2 on the previous line. Or you could have ?x2, 2x2, 2x?. But that means you would have (?+2)x2 on the previous line, and so you've also described it wrong.

I have just mentioned 'if you start from "1"'. You could start with other sequences. What happens if you start with "22"?