Liars and Truth Tellers

Consider the guy at the very left. He has no liars to his left, so the statement he speaks must be wrong. Hence, he is a liar. Now consider the guy at the very right. He has no truth tellers to his right but at least one liar to his left, so his statement must be true, so he must be a truth teller. Now consider the guy just to the right of the leftmost guy. He has one liar to his left and at least one truth teller to his right, so his statement is false, so he must be a liar. Again, consider the guy just to the left of the rightmost guy. He has at least two liars to his left but only one truth teller to his right, so his statement is true, implying he is a truth teller. We do this throughout the whole series of people; eventually we find out that the first $\dfrac{2014}{2} = 1007$ people (from the left) are liars and the rest are truth tellers. Hence, our answer is $\boxed{1007}$ .

To pick a grammar nit, the men should say: "There are (strictly) FEWER truth tellers to my right than liars to my left." Truth tellers and liars are countable, so use "fewer" not "less".

It is simple to think that for all truth tellers to be truthful, they always have to have more of liars to left and less of truth tellers to right, which implies that all of truth tellers are together and all liars together.

Now, we know that truth tellers are to the right and liars to the left, for statements to match.

So for the rightmost liar, too, his statement has to be false, which implies that he has to be at the 1007th position. This is so to make the statement of the truth teller to his right true.

Generally speaking, one can prove the following:

Suppose that there are $N$ people standing on a straight line. $N$ is a positive integer (or a natural number).

(Assume that each people is either a liar (who only tell lies) or a truth teller (who only tells the truth.))

If each of the $N$ people says "there are (strictly) FEWER truth tellers to my right than liars to my left" or "the number of truth tellers to my right is (strictly) LESS than the number of liars to my left", then the number of liars is $\lceil{N/2}\rceil$ .

There are an even number of men (2014) standing in the line. So if there are an equal number of liars to truth-tellers, just halve the number of men to find how many liars or truth-tellers there are (1007).