Difference between revisions of "2007 AMC 10A Problems/Problem 11"
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== Solution == | == Solution == | ||
The sum of the numbers on the top face of a cube is equal to the sum of the numbers on the bottom face of the cube; these <math>8</math> numbers represent all of the vertices of the cube. Thus the answer is <math>\frac{1 + 2 + \cdots + 8}{4} = 18\ \mathrm{(C)}</math>. | The sum of the numbers on the top face of a cube is equal to the sum of the numbers on the bottom face of the cube; these <math>8</math> numbers represent all of the vertices of the cube. Thus the answer is <math>\frac{1 + 2 + \cdots + 8}{4} = 18\ \mathrm{(C)}</math>. | ||
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| + | == Solution 2 == | ||
| + | Consider a number on a vertex. It will be counted in 3 different faces, the ones it is on. Therefore, each number <math>1,2,\cdots,7,8</math> will be added into the total sum <math>3</math> times. Therefore, our total sum is <math>3(1+2+\cdots+8)=108.</math> Finally, since there are <math>6</math> faces, our common sum is <math>\dfrac{108}{6}=\mathrm{(C) } 18.</math> | ||
== See also == | == See also == | ||
Revision as of 16:22, 2 December 2015
Contents
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 the top face of a cube is equal to the sum of the numbers on the bottom 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, the ones it is on. 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 
See also
| 2007 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 10 |
Followed by Problem 12 | |
| 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. 
