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Difference between revisions of "2016 AMC 10A Problems/Problem 18"

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<math>\textbf{(A) } 1\qquad\textbf{(B) } 3\qquad\textbf{(C) }6 \qquad\textbf{(D) }12 \qquad\textbf{(E) }24</math>
 
<math>\textbf{(A) } 1\qquad\textbf{(B) } 3\qquad\textbf{(C) }6 \qquad\textbf{(D) }12 \qquad\textbf{(E) }24</math>
  
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===Solution 1===
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==Solution 1==
  
 
First of all, the adjacent faces have same sum <math>(18</math>, because <math>1+2+3+4+5+6+7+8=36</math>, <math>36/2=18)</math>,  
 
First of all, the adjacent faces have same sum <math>(18</math>, because <math>1+2+3+4+5+6+7+8=36</math>, <math>36/2=18)</math>,  
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Now, the problem is same as the problem to arrange <math>4</math> points in a <math>2-D</math> square. which is <math>4!/4</math>=<math>\boxed{\textbf{(C) }6.}</math>
 
Now, the problem is same as the problem to arrange <math>4</math> points in a <math>2-D</math> square. which is <math>4!/4</math>=<math>\boxed{\textbf{(C) }6.}</math>
  
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=== Solution 2 ===
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== Solution 2 ==
  
 
Again, all faces sum to <math>18.</math> If <math>x,y,z</math> are the vertices next to one, then the remaining vertices are <math>17-x-y, 17-y-z, 17-x-z, x+y+z-16.</math> Now it remains to test possibilities. Note that we must have <math>x+y+z>17.</math> WLOG let <math>x<y<z.</math>
 
Again, all faces sum to <math>18.</math> If <math>x,y,z</math> are the vertices next to one, then the remaining vertices are <math>17-x-y, 17-y-z, 17-x-z, x+y+z-16.</math> Now it remains to test possibilities. Note that we must have <math>x+y+z>17.</math> WLOG let <math>x<y<z.</math>
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So our answer is <math>3\cdot 2=\boxed{\textbf{(C) }6.}</math>
 
So our answer is <math>3\cdot 2=\boxed{\textbf{(C) }6.}</math>
  
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=== Solution 3 ===
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== Solution 3 ==
  
 
We know the sum of each face is <math>18.</math> If we look at an edge of the cube whose numbers sum to <math>x</math>, it must be possible to achieve the sum <math>18-x</math> in two distinct ways, looking at the two faces which contain the edge. If <math>8</math> and <math>6</math> were on the same face, it is possible to achieve the desired sum only with the numbers <math>1</math> and <math>3</math> since the values must be distinct. Similarly, if <math>8</math> and <math>7</math> were on the same face, the only way to get the sum is with <math>1</math> and <math>2</math>. This means that <math>6</math> and <math>7</math> are not on the same edge as <math>8</math>, or in other words they are diagonally across from it on the same face, or on the other end of the cube.
 
We know the sum of each face is <math>18.</math> If we look at an edge of the cube whose numbers sum to <math>x</math>, it must be possible to achieve the sum <math>18-x</math> in two distinct ways, looking at the two faces which contain the edge. If <math>8</math> and <math>6</math> were on the same face, it is possible to achieve the desired sum only with the numbers <math>1</math> and <math>3</math> since the values must be distinct. Similarly, if <math>8</math> and <math>7</math> were on the same face, the only way to get the sum is with <math>1</math> and <math>2</math>. This means that <math>6</math> and <math>7</math> are not on the same edge as <math>8</math>, or in other words they are diagonally across from it on the same face, or on the other end of the cube.

Revision as of 22:06, 22 November 2016

Problem

Each vertex of a cube is to be labeled with an integer $1$ through $8$, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How many different arrangements are possible?

$\textbf{(A) } 1\qquad\textbf{(B) } 3\qquad\textbf{(C) }6 \qquad\textbf{(D) }12 \qquad\textbf{(E) }24$

Solution 1

First of all, the adjacent faces have same sum $(18$, because $1+2+3+4+5+6+7+8=36$, $36/2=18)$, consider the $opposite \text{ } sides$ (the two sides which are parallel but not in same face of the cube) they must have same sum value too. Now think about the extreme condition 1 and 8 , if they are not sharing the same side, which means they would become end points of $opposite \text{ } sides$, we should have $1+X=8+Y$, but no solution for $[2,7]$, contradiction.

Now we know $1$ and $8$ must share the same side, which sum is $9$, the $opposite \text{ } side$ also must have sum of $9$, same thing for the other two parallel sides.

Now we have $4$ parallel sides $1-8, 2-7, 3-6, 4-5$. thinking about $4$ end points number need to have sum of $18$. it is easy to notice only $1-7-6-4$ vs $8-2-3-5$ would work.

So if we fix one direction $1-8 ($or $8-1)$ all other $3$ parallel sides must lay in one particular direction. $(1-8,7-2,6-3,4-5)$ or $(8-1,2-7,3-6,5-4)$

Now, the problem is same as the problem to arrange $4$ points in a $2-D$ square. which is $4!/4$=$\boxed{\textbf{(C) }6.}$

Solution 2

Again, all faces sum to $18.$ If $x,y,z$ are the vertices next to one, then the remaining vertices are $17-x-y, 17-y-z, 17-x-z, x+y+z-16.$ Now it remains to test possibilities. Note that we must have $x+y+z>17.$ WLOG let $x<y<z.$

$3,7,8:$ Does not work. $4,6,8:$ Works. $5,6,7:$ Does not work. $5,6,8:$ Works. $5,7,8:$ Does not work. $6,7,8:$ Works.

So our answer is $3\cdot 2=\boxed{\textbf{(C) }6.}$

Solution 3

We know the sum of each face is $18.$ If we look at an edge of the cube whose numbers sum to $x$, it must be possible to achieve the sum $18-x$ in two distinct ways, looking at the two faces which contain the edge. If $8$ and $6$ were on the same face, it is possible to achieve the desired sum only with the numbers $1$ and $3$ since the values must be distinct. Similarly, if $8$ and $7$ were on the same face, the only way to get the sum is with $1$ and $2$. This means that $6$ and $7$ are not on the same edge as $8$, or in other words they are diagonally across from it on the same face, or on the other end of the cube.

Now we look at three cases, each yielding two solutions which are reflections of each other:

1) $6$ and $7$ are diagonally opposite $8$ on the same face. 2) $6$ is diagonally across the cube from $8$, while $7$ is diagonally across from $8$ on the same face. 3) $7$ is diagonally across the cube from $8$, while $6$ is diagonally across from $8$ on the same face.

This means the answer is $3\cdot 2=\boxed{\textbf{(C) }6.}$

See Also

2016 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 17 		Followed by
Problem 19
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
2016 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 13 		Followed by
Problem 15
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 12 Problems and Solutions

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