Sphere-packing problems are difficult and beyond the scope of this exam.
I'll bet the OA is uphillclimb's answer, but it isn't obvious. The problem is that the lattice there is a very bad packing and you can change the lattice at least along the sides to a close-packing lattice. Particularly around the corners, it is not at all clear that you can't fit more balls in than this. This problem is messed.
BTW - sphere packing is a problem that interested Gauss, Kepler, and oodles of very fine mathematicians.
Thanks for the info on these great mathematicians....
Yes the wording should be very good even to attempt these questions