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| Math Central | Quandaries & Queries |
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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
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