In each of the eight vertices of the cube, a nonzero number was written, and on each face

In each of the eight vertices of the cube, a nonzero number was written, and on each face – the product of four numbers located at its vertices. What is the largest number of negative numbers among all 14 written numbers?

since the number on a face is the result of the product of numbers on four adjacent vertices, it is maximum possible to write 3 negative numbers on the vertices of each face, because the product of an odd number of negative numbers is also negative. Since some vertices of the cube are common for three faces at once (and there are 6 of them in total), it will be enough to sign positive numbers for only two vertices. Thus, we subtract 2 positive numbers from the total number of numbers: 14-2 = 12.
The correct answer is 12.