fish2xs ~
"Slicing the pie" refers to two things. First, it refers to the size of the area to be searched at one time. Second (and more importantly), it refers to the shape of it.
First, regarding size. Just as you would not eat an entire pie at one time, it is bad tactics to search a whole room at once. It is easier to thoroughly search a small area than it is to thoroughly search a larger one. So you need to cut it down into manageable pieces so that nothing is missed. That's "slicing the pie" into reasonable pieces.
The chunks that you break your room into are pie-shaped. Visualize yourself standing just outside the doorway, a couple feet back from the entrance and off to one side so that you cannot be seen from the room you are going to enter. From this spot, you can't see much of the room -- just a slim triangle-shaped "slice" of the room and that's where you look first. Make sure there is nothing at all in that pie-shaped wedge. When that is clear, you move slightly so that the visible wedge gets a little wider, and search the newly visible area. Keep doing that (without entering the room), until you have seen as much of the room as can be seen from the doorway.
Eventually you will have to take a leap into most rooms. There will be a small, pie-shaped segment of the room you haven't seen yet. If you were on the right side of the doorway, the unsearched segment will be where the doorway wall meets the right wall of the room. As you enter the room, you need to "pie" that corner.
Words are pretty poor for this sort of thing and I'm sure someone else could explain it better. Anyone got a sketch?
pax