3. [5] 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?
Answer: 168 Assign coordinates in such a way that the dragoon being fired on is centered at $(0,0)$. Any dragoon firing at it must have a center with $x$-coordinates and $y$-coordinates that are no smaller than -6 and no greater than 6 . That means that every dragoon firing at it must lie entirely in the region bounded by the lines $x=-6.5, x=6.5, y=-6.5$, and $y=6.5$. This is a square with sides of length 13 , so there is room for exactly 169 dragoons in it. One of them is the dragoon being fired on, so there are at most 168 dragoons firing at it.