# Example of a superword w such that v^2 isn't its subword

What is an example of an infinite word(superword) w such that if a nonempty word v belongs to L = {1,2,3}*, v^2 isn't a subword of w?

For example if w = 123123123...123 and v = 123, v^2 = 123123 hence it's a subword of w, I can't seem to find a superword that fits the requirement.

• I think 123111222333111112222233333... works? No? – Pavel Nov 15 '14 at 18:34
• at first we have one 1, one 2, one 3, then three 1's, three 2's and three 3's, then 5 of each, then 7 of each, then 9 of each, etc – Pavel Nov 15 '14 at 18:35
• But you have double letters - the word "1" repeats a lot as $1^2$ (same goes for 2 and 3) – Shaull Nov 15 '14 at 18:38
• oh yeah, you're right, hmm, seems impossible – Pavel Nov 15 '14 at 18:39
• – Raphael Nov 16 '14 at 13:36