At each vertex of a cube we write either 1 or 1 Then in the

At each vertex of a cube we write either 1 or 1. Then in the middle of each face we write the product of the numbers at the vertices. Can the sum of the obtained 14 numbers be zero? Hint: Find the product of these 14 numbers.

Solution

No,

For the sum of fourteen + and - 1s to equal zero, seven will have to be +, and seven -.

Note that situation requires an ODD number of each!

Now start with all vertices assigned +1; then all faces will also be assigned 1 for a total of 14.

If we change the sign of a vertex one at a time and note that the vertex and its three adjacent vertices CHANGE sign.

After a while a vertex might be surrounded by 0, 1, 2, or 3 negative faces. Check in each case how the number of negative 1s goes up or down. For example, if the vertex goes from + to -, and at that time just one of the adjacent faces is -, then three items will go from + to - and one goes from - to +.

In all we will end up with two more -\'s and two fewer +\'s --

In other words, the numbers change by EVEN increments.

hence 7 \"+1\" & 7 \"-1\" can not be possible.