Question about turing machines
The Turing machine uses the tape alphabet $\Gamma = \{0,1,x,\epsilon\}$.
Writing down all the transitions explicitly is tedious, so I'll just summarize the main steps:


Move right until you find a 1 or $\epsilon$.
If a 1 is found proceed as follows:


Replace the $1$ with $x$.
Move the head back to the initial position (move left until you find $\epsilon$, then move right once).
Move right until you find a $0$ or $\epsilon$. If no $0$ is found reject.
Replace the 0 with $x$.
Move right until you find a $0$ or $\epsilon$. If no $0$ is found reject.
Replace the 0 with $x$.
Move the head back to the initial position.
Repeat everything from the very first step.

Otherwise, if no 1 is found:


Move left until you find a $0$ or $\epsilon$. If no 0 is found, accept. 
Reject.