2023 AMC 10A Problems/Problem 6
Contents
Problem
An integer is assigned to each vertex of a cube. The value of an edge is defined to be the sum of the values of the two vertices it touches, and the value of a face is defined to be the sum of the values of the four edges surrounding it. The value of the cube is defined as the sum of the values of its six faces. Suppose the sum of the integers assigned to the vertices is . What is the value of the cube?
Solution 1
Each of the vertices is counted times because each vertex is shared by three different edges. Each of the edges is counted times because each edge is shared by two different faces. Since the sum of the integers assigned to all vertices is , the final answer is
~Mintylemon66
Solution 2
Note that each vertex is counted times. Thus, the answer is .
~Mathkiddie
Solution 3
Just set one vertex equal to , it is trivial to see that there are faces with value , and .
~SirAppel ~minor edits by TiguhBabeHwo
Solution 4
Since there are 8 vertices in a cube, there are vertices for two edges. There are edges per face, and faces in a cube, so the value of the cube is .
~DRBStudent ~Failure.net
(Minor formatting by Technodoggo)
Solution 5 (use an example)
Set each vertex to value 1, so the sum of the vertices is 8. We find that the value of the cube, if all vertices are 1, is 48. We conclude that the value of the cube is 6 times the value of the sum of the vertices. Therefore, we choose
~milquetoast
Solution 6
The wording of the problem implies that the answer should hold for any valid combination of integers. Thus, we choose the numbers , which are indeed integers that add to . Doing this, we find three edges that have a value of , and from there, we get three faces with a value of (while the other three faces have a value of ). Adding the three faces together, we get .
~MathHafiz
Video Solution by Math-X (First understand the problem!!!)
https://youtu.be/GP-DYudh5qU?si=3nDWUtwfk6n3ka9V&t=1240 ~Math-X
Video Solution by CosineMethod [🔥Fast and Easy🔥]
https://www.youtube.com/watch?v=9KVpfLeW9ro
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Video Solution (easy to digest) by Power Solve
See Also
2023 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.