Difference between revisions of "2003 AMC 12A Problems/Problem 13"

m
(Solution)
Line 8: Line 8:
 
== Solution ==
 
== Solution ==
 
{{image}}
 
{{image}}
 +
[[Image:2003amc10a10.gif]]
  
 
Let the squares be labeled <math>A</math>, <math>B</math>, <math>C</math>, and <math>D</math>.
 
Let the squares be labeled <math>A</math>, <math>B</math>, <math>C</math>, and <math>D</math>.
Line 19: Line 20:
 
Squares  <math>4</math>, <math>5</math>, <math>6</math>, <math>7</math>, <math>8</math>, and <math>9</math> will allow the polygon to become a cube with one face missing when folded.  
 
Squares  <math>4</math>, <math>5</math>, <math>6</math>, <math>7</math>, <math>8</math>, and <math>9</math> will allow the polygon to become a cube with one face missing when folded.  
  
Thus the answer is <math>6 \Rightarrow E</math>.  
+
Thus the answer is <math>6 \Rightarrow E</math>.
  
 
== See Also ==
 
== See Also ==

Revision as of 15:24, 23 March 2010

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


An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.


2003amc10a10.gif

Let the squares be labeled $A$, $B$, $C$, and $D$.

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