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Difference between revisions of "2003 AMC 12A Problems/Problem 13"
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== Problem ==
 
== Problem ==
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The polygon enclosed by the solid lines in the figure consists of 4 congruent squares joined edge-to-edge. One more congruent square is attatched to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face missing?  
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The [[polygon]] enclosed by the solid lines in the figure consists of 4 [[congruent]] [[square (geometry) | squares]] joined [[edge]]-to-edge. One more congruent square is attatched to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a [[cube (geometry) | cube]] with one face missing?  
  
 
[[Image:2003amc10a10.gif]]
 
[[Image:2003amc10a10.gif]]
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== Solution ==
 
== Solution ==
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Let the squares be labeled <math>A</math>, <math>B</math>, <math>C</math>, and <math>D</math>.
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Let the squares be labeled <math>A</math>, <math>B</math>, <math>C</math>, and <math>D</math>. {{image}}
  
 
When the polygon is folded, the "right" edge of square <math>A</math> becomes adjacent to the "bottom edge" of square <math>C</math>, and the "bottom" edge of square <math>A</math> becomes adjacent to the "bottom" edge of square <math>D</math>.  
 
When the polygon is folded, the "right" edge of square <math>A</math> becomes adjacent to the "bottom edge" of square <math>C</math>, and the "bottom" edge of square <math>A</math> becomes adjacent to the "bottom" edge of square <math>D</math>.  

Revision as of 10:24, 11 November 2006

Problem

The polygon enclosed by the solid lines in the figure consists of 4 congruent squares joined edge-to-edge. One more congruent square is attatched to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face missing?

2003amc10a10.gif

$\mathrm{(A) \ } 2\qquad \mathrm{(B) \ } 3\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 5\qquad \mathrm{(E) \ } 6$

Solution

Let the squares be labeled $A$, $B$, $C$, and $D$. ---- An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.

When the polygon is folded, the "right" edge of square $A$ becomes adjacent to the "bottom edge" of square $C$, and the "bottom" edge of square $A$ becomes adjacent to the "bottom" edge of square $D$.

So, any "new" square that is attatched to those edges will prevent the polygon from becoming a cube with one face missing.

Therefore, squares $1$, $2$, and $3$ will prevent the polygon from becoming a cube with one face missing.

Squares $4$, $5$, $6$, $7$, $8$, and $9$ will allow the polygon to become a cube with one face missing when folded.

Thus the answer is $6 \Rightarrow E$.

See Also

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