An ant is a methodical creature, and the ants in this puzzle are particularly so. When they start walking they always walk in a straight line, and when they reach a boundary they always turn exactly $90^\circ$ anticlockwise. If they still cannot move, they turn through $180^\circ$ and try to move. After this they have tried all directions and so stop. (Put otherwise, the ant tries turning left; if that fails, it tries turning right from its original heading.)
When an ant is set on the lower left corner on a $5\times 5$ board, as shown below as cell $A1$, and it treats any cell it has already visited, as well as the edges of the board, as boundaries, it traverses the entire board (as shown in blue).
Given the $11\times 11$ grid below, there are two starting cells for such an ant that omit exactly one cell when the ant cannot move any more. The ant always starts moving the $A \rightarrow K$ direction, unless it starts in column $K$ in which case, applying the $90^\circ$ rule, it moves up. Which two cells are they?
EDIT: As pointed out by Stiv there is actually only one starting cell for which the ant will omit a single cell, not two.