We build a $4 \times 4 \times 4$ cube from sugar cubes. How many different rectangular prisms are determined by the sugar cubes, if the prisms differ in at least one sugar cube?
I. solution. A cuboid can be uniquely determined within the cube by taking a pair of planes parallel to each of the cube's different pairs of faces, which pass through the appropriate vertices of the small sugar cubes. Since the large cube is $4 \times 4 \times 4$, 5 different suitable planes can be taken parallel to a given pair of faces; thus, a pair of planes can be chosen in $\binom{5}{2}=10$ ways. The cube has three different pairs of faces, and a pair of planes can be chosen independently for each of these, so in total $10^{3}=1000$ different cuboids are defined by the sugar cubes forming the $4 \times 4 \times 4$ cube.

II. solution. Place the $4 \times 4 \times 4$ cube in a three-dimensional coordinate system so that its vertices are the points $A(0 ; 0 ; 0), B(4 ; 0 ; 0), C(4 ; 4 ; 0), D(0 ; 4 ; 0), E(0 ; 0 ; 4), F(4 ; 0 ; 4), G(4 ; 4 ; 4)$ and $H(0 ; 4 ; 4)$. Then a cuboid can be defined by the two endpoints of a space diagonal. Let's count how many space diagonal vectors can be taken (in this case, two vectors are considered different if their starting and ending points do not match, so even vectors of the same length and direction are considered different in the given case). A vector will be a space diagonal vector if and only if none of the coordinates of its starting and ending points match. The coordinates of the points we are examining can only be $0,1,2,3$ or 4, so we are examining a total of $5^{3}=125$ points. To a given point, $4^{3}=64$ suitable points can be assigned that form a space diagonal vector with it. Therefore, a total of $125 \cdot 64=8000$ different space diagonal vectors have been taken. Each cuboid has 4 space diagonals, and each of these can be taken as a vector in two directions; thus, we counted each cuboid 8 times. Therefore, a total of $\frac{8000}{8}=1000$ different cuboids, which meet the requirements of the problem, are defined by the sugar cubes.