If you continue generating the numbers to this list, do numerals greater than 3 ever appear? If so, how far do you have to go for a 4 to appear? And a 5 and 6, and so on?
(I am purposely not explaining how to generate the next number in the list because if you DON’T know how to make the next entry, I don’t want to take away the joy of discovering it.)
1
11
21
1211
111221
312211
13112221
. . . .
No. As you know ( if you know how the sequence is generated) the numbers are formed in pairs. Suppose the number 4 appears somewhere and consider the first row in which it appears. The the previous row must have had four consecutive identical digits, say NNNN. Ther are two possibilities:
(i) xN NN Ny
This is impossible; since the xN and NN would actually have been combined into (x+N)N
(ii)NN NN
This is impossible for the same reason.
Thanks Jabba.