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−  == Problem ==
 +  #REDIRECT[[2003 AMC 12A Problems/Problem 13]] 
−  The polygon enclosed by the solid lines in the figure consists of 4 congruent squares joined edgetoedge. One more congruent square is attached 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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−  [[Image:2003amc10a10.gif]]  
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−  <math> \mathrm{(A) \ } 2\qquad \mathrm{(B) \ } 3\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 5\qquad \mathrm{(E) \ } 6 </math>
 
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−  == Solution ==
 
−  [[Image:2003amc10a10solution.gif]]
 
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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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−  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>.
 
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−  So, any "new" square that is attached to those edges will prevent the polygon from becoming a cube with one face missing.
 
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−  Therefore, squares <math>1</math>, <math>2</math>, and <math>3</math> will prevent the polygon from becoming a cube with one face missing.
 
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−  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.
 
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−  Thus the answer is <math>6 \Rightarrow E</math>.
 
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−  Another way to think of it is that a cube missing one face has <math>5</math> of it's <math>6</math> faces. Since the shape has <math>4</math> faces already, we need another face. The only way to add anopther face is if the added square does not overlap any of the others. <math>1</math>,<math>2</math>, and <math>3</math> overlap, while <math>4 \Rightarrow</math> 9 do not. The answer is <math>6 \Rightarrow E</math>
 
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−  == See Also ==
 
−  {{AMC10 boxyear=2003ab=Anumb=9numa=11}}
 
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−  [[Category:Introductory Geometry Problems]]
 