You have to subtract out the number of fractional dies touching the circle (on the edge of the wafer). The cirumference is \[\pi\]*wafer diameter, the width of each partial die is at most (when at a 45\[^{o}\]angle), \[\sqrt{{width}^2+{width}^2}\]. The ratio is the number that fit on the circle, on average, approximately.
Now you ask why the "most" case? Won't many of the times the dies be on top of each other, so why the longest part of the die? I guess the reason is because in the "least" case, where the width alone is the packing space on the edge of the circle, there is not much of a cut off, if you line it up correctly. (You don't want the central die necessarily at the center of the circle.) So you can beat out a random placement slightly.
Jason