I can't figure out how to swap boxes on a Turing Machine tape.
So for example, I have a tape that says
a 1 0 1 1 1 0
^
And I want to move that a over so that it is in the middle. (but my end goal is not important here)
How do I get the Turing Machine to do
1 a 0 1 1 1 0
^
1 0 a 1 1 1 0
^
etc...
Here is a possible set of transition to do one move, as you request.
Note that $x$ is to be replaced by $0$ and by $1$, so that each
transition with $x$ is actually a pattern to be replaced by 2
transitions, one with $0$ and the other with $1$, instead of $x$ (so you really have 5 transitions below)
And of course $q_{2,x}$ stands for two states: $q_{2,0}$ and
$q_{2,1}$.
I present things that way so that you know what to do if you can have more than two disctinct symbols to deal with.
The $*$ is just a wild card, indicating that the content of the tape does not matter.
This will do the required exchange, with $0$ or $1$. You end up in state $q_2$, still pointing to the $a$
$\begin{align} q_1, a&\to q_{1,a},a,R\\ q_{1,a},x&\to q_{2,x},a,L\\ q_{2,x},*&\to q_2,x,R \end{align}$
I hope this is clear.
What it does is to note $a$ in the finite state control, by going in state $q_{1,a}$, and then moving Right. There it notes in the finite state control whatever $x$ is found there, a $0$ or a $1$, thus going in state $q_{2,x}$, and write down the $a$ (so it can forget it), and moves back Left. Back on the left square, there is still an $a$, it does not matter, it can write down the $x$ it memorized in the finite state control, and move Right again, in state $q_2$, so that it is pointing to the $a$ that was moved.
Note that the $*$ could be replaced by $a$. The $*$ is a convenient notation to say that it does not matter, as you could have several symbols other than $a$ to be moved in this way.
$q_2$ is whatever state you need to be in after performing one such permutation.
To this end, you first need to find the middle point. In this answer there is a good algorithms to do that:
Then, you move the "a" until you get to the marker, and remove the marker.
EDIT:
How to move a letter: Say you have an "a" among a string of 0 and 1. here is a way to shift it to the right:
where $*$ is "don't care".
What is going on here? If we see an $a$, we move to the cell to the right of it, remember what was written there, and write an $a$ instead. If it was a $0$ we deleted, we move to state $q_0$, if it was a $1$ we move to $q_1$. This is how we "remember" what was deleted. Then we move the head one step to the left, and write 0 or 1 according to whether we are in state $q_0$ or $q_1$. Then we move one step to the right and repeat the same process. Try to simulate it for yourself on a paper...
