Designing a DFA that accepts strings such that nth character from last satisfies condition This is a homework question, so I am only looking for hints.

I got a question in an assignment which states :


  Design a DFA that accepts strings having 1 as the 4th character from the end, on the alphabet {0,1}


I have been at this for a few hours now, and I think that designing such a DFA is not possible. However, I am not sure how to move forward in this direction to write up a somewhat formal proof. 

So, what should I try to do to prove or disprove my hypothesis?
Hint: Your language can be described by the regular expression $\{0,1\}^*1\{0,1\}^3$.

Another hint: Your DFA can remember the last $4$ characters (and maintain that information). You will also need states to handle the case in which less than $4$ characters have been seen so far.