OK, so this is what I got.
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Create an infinite grid of copies of the room by mirroring the room, with the original room in the center. Each copy gets a copy of Ax (but not of Larry). Now there's a 1:1 correspondence between the possible attacks and the straight lines from Ax copies to Larry.
Consider the set of midpoints of those straight lines, reflected back to the original rectangle.
They form 16 distinct points. Why only 16? It turns out that if you move by 4 rooms on the infinite grid in any direction, the midpoint (reflected back) is the same. Consider for instance the grid at (0,0) (the original room) vs (4,0). The midpoint lies in the room at (2,0), which has the same layout as the original room (because it was mirrored twice). That's why situation from (4,0) is basically like a magnified version of the original situation at (0,0). A similar argument can be made for moving by 4 rooms in any other direction. Hence it is sufficient to consider any 4x4 square of rooms.
If you are interested, I wrote a few lines of code to compute the coordinates that I'm happy to share.
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