I read about the Champernowne constant on Wikipedia a couple of days ago, and I got curious about something similar: is there some "Champernowne-like" number; that is, a concatenation of all numbers up to some $n \ge 2$ (like $1234567891011$), that is a perfect square? I've done a computer search up to $n=2000$, but I haven't found any.
Are there any? If not, how may we prove this? Any thoughts on this are welcome!