7.2. Is it possible to place the 12 numbers $1,2, \ldots, 12$ on the edges of a cube such that the product of the four numbers on the top face equals the product of the four numbers on the bottom face?
Answer: It is possible. Solution. An example of such an arrangement can be as follows: on the top face, place the numbers $2,4,9,10$; on the bottom face, place the numbers $3,5,6,8$; the remaining numbers $1,7,11,12$ are placed on the side edges. The products on the top and bottom faces are the same and equal to 720. (To arrive at such an example, one needs to select the prime numbers 7 and 11 for the side edges and distribute the powers of two and three on the top and bottom faces, which significantly reduces the number of combinations to check.)