SEARCH HOME
 Math Central Quandaries & Queries
 Question from Shankar, a teacher: Hello Sir Can you pls tell me, how to solve this problem ? Every vertex of a cube is assigned a number +1 or -1. Every face has a number that is the product of all the numbers in its corners. Then the 14 numbers are summed up ( all the vertices and faces ). Can the sum be 0? Thanks a lot !

Shankar,

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. 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. You should be able to see how the argument goes from here.

Chris

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.