1985 AJHSME Problems/Problem 11

Problem

A piece of paper containing six joined squares labeled as shown in the diagram is folded along the edges of the squares to form a cube. The label of the face opposite the face labeled $\text{X}$ is

$[asy] draw((0,0)--(0,1)--(2,1)--(2,2)--(3,2)--(3,0)--(2,0)--(2,-2)--(1,-2)--(1,0)--cycle); draw((1,0)--(1,1)); draw((2,0)--(2,1)); draw((1,0)--(2,0)); draw((1,-1)--(2,-1)); draw((2,1)--(3,1)); label("U",(.5,.3),N); label("V",(1.5,.3),N); label("W",(2.5,.3),N); label("X",(1.5,-.7),N); label("Y",(2.5,1.3),N); label("Z",(1.5,-1.7),N); [/asy]$

$\text{(A)}\ \text{Z} \qquad \text{(B)}\ \text{U} \qquad \text{(C)}\ \text{V} \qquad \text{(D)}\ \ \text{W} \qquad \text{(E)}\ \text{Y}$

Solution

To find the face opposite $\text{X}$ , we can find the faces sharing an edge with $\text{X}$ , so the only face remaining will be the opposite face.

Clearly, $\text{V}$ and $\text{Z}$ share an edge with $\text{X}$ . Also, the faces $\text{V}$ , $\text{X}$ , and $\text{W}$ share a common vertex, therefore $\text{X}$ shares an edge with $\text{W}$ . Similarly, the faces $\text{U}$ , $\text{V}$ , and $\text{X}$ share a common vertex, so $\text{X}$ shares an edge with $\text{U}$ .

The only face $\text{X}$ doesn't share an edge with is $\text{Y}$ , which is choice $\boxed{\text{D}}$

1985 AJHSME (Problems • Answer Key • Resources)
Preceded by Problem 10	Followed by Problem 12
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1985 AJHSME Problems/Problem 11

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