If your room is surrounded in all directions by other rooms, you share one sphere with 7 other rooms. (top 4 and bottom 4)
Thus a single particle gets shared by 8 rooms. The part in your room is 1/8 And the number of such particles are 8 (as cubic room has 8 vertices)
So in CCP the space occupied by all eight particles is basically equal to that of a single particle.
Now as assumed in the question, the particles are tightly packed, the Radius of sphere equals half of the edge length. In other words edge length is twice the radius of the sphere.Packingefficency=Volume of sphereVolume of cubePackingefficency=Volume of sphereVolume of cube=z⋅43πr3(2r)3=z⋅(Herez is net particle contribution; for the current example it is equal to 1)
Calculating this you'll get packing efficiency≈52.4 (for ccp)
for bcc its 32%
