Consider a $n \times n$ square grid (finite) (a square is divided into smaller squares by lines parallel to its sides). The boundary of the square is oriented, (clockwise or anticlockwise) that is, a direction is chosen on it and fixed, such that if you move in that direction along the boundary, the internal points of the square always stay on your left or on your right (depending on the orientation). For each of the internal edges of the subdivision, a direction is specified, such that for each interior vertex, there are exactly two edges coming to the vertex and two edges going away from it (see diagram below).
Then my question is that does it follow that there is atleast one oriented face in the subdivision?
(For example in the figure, there is exactly one such face, namely in the extreme lower right corner).