2007 AMC 10A Problems/Problem 11
Problem
The numbers from to are placed at the vertices of a cube in such a manner that the sum of the four numbers on each face is the same. What is this common sum?
Solution
The sum of the numbers on one face of the cube is equal to the sum of the numbers on the opposite face of the cube; these numbers represent all of the vertices of the cube. Thus the answer is .
Solution 2
Consider a number on a vertex. It will be counted in 3 different faces, since any vertex is the intersection of three edges. Therefore, each number will be added into the total sum times. Therefore, our total sum is Finally, since there are faces, our common sum is
Video Solution
https://youtu.be/ZhAZ1oPe5Ds?t=2075
~ pi_is_3.14
See also
2007 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
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All AMC 10 Problems and Solutions |
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