These are not the droids that you are looking for - Part 1

Maybe this might help you. We can assign binary truth values to the statements to get the right case. We'll form a truth table with one column of truth values of the three statements for each droid and then choose the column (droid) which matches the criteria of our problem (exactly one truth).

$\begin{array}{|l|c|c|c|}\hline &\textrm{Droid 1}&\textrm{Droid 2}&\textrm{Droid 3}\\ \hline \textrm{Droid 1's Message}&1&0&0\\~\\ \textrm{Droid 2's Message}&0&1&0\\~\\ \textrm{Droid 3's Message}&1&0&0\\ \hline\end{array}$

From the column of Droid 2, it is perfectly clear that Droid 2 is the answer since only that case ensures exactly one true statement.

By the process of elimination by looking for contradiction. As mentioned in my solution. If we assume Droid 3's statement is true, then there would be two true statements which contradict the given fact that there is only one true statement. Therefore Droid 3's statement must be false. Now that we know that Droid 3's statement is false this means the Droid 1 is not the droid. And Droid 1's statement is therefore false. Now we have two false statement, therefore, the statement of Droid 2 must be true. You can also start with assuming Droid 1's statement is true, then Droid 3's statement is also true, which again contradict. Therefore, the Droid 2's statement is true, and Droid 2 is the droid. We can't start with Droid 2's statement because it has no other information to refer to.