So, I know that any multiple-tape TM can be in theory turned into a one-tape TM. However, it is too easy to copy, let's say, binary numbers from one tape to another. That's why I thought about putting a separator between the two copies and then taking symbols one after the other and writing them after the separator until the separator itself is encountered. The problem, however, is that I am not sure how it could remember places where it has already been/copied the characters. Example:
First we have:
Then we put the separator '&' at the end
Read back to beginning and change the state accordingly so that it will write $1$ or $0$ after the separator. So far, so good, then we read back again, and now:
How could we know that we have already copied the first $1$ and must now copy the $0$ without putting any restriction on the input length (in regards to the number of states)? In other words, how could we remember the last copied symbol?
I have thought of putting an extra parameter - just a integer ≤ length (something like $\delta (z_1,1,L,1)$ where the last one would be the number of already written-out symbols). This would be easy to understand, but would be nowhere near the definition of Turing machine. So, any useful ideas?