HMMT November 2010 F3

Dragoons take up $1 \times 1$ squares in the plane with sides parallel to the coordinate axes such that the interiors of the squares do not intersect. A dragoon can fire at another dragoon if the difference in the $x$-coordinates of their centers and the difference in the $y$-coordinates of their centers are both at most 6 , regardless of any dragoons in between. For example, a dragoon centered at $(4,5)$ can fire at a dragoon centered at the origin, but a dragoon centered at $(7,0)$ can not. A dragoon cannot fire at itself. What is the maximum number of dragoons that can fire at a single dragoon simultaneously?