|
|
| Line 1: |
Line 1: |
| − | == Problem ==
| + | #REDIRECT[[2003 AMC 12A Problems/Problem 13]] |
| − | The polygon enclosed by the solid lines in the figure consists of 4 congruent squares joined edge-to-edge. 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?
| |
| − | | |
| − | [[Image:2003amc10a10.gif]] | |
| − | | |
| − | <math> \mathrm{(A) \ } 2\qquad \mathrm{(B) \ } 3\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 5\qquad \mathrm{(E) \ } 6 </math>
| |
| − | | |
| − | == Solution ==
| |
| − | [[Image:2003amc10a10solution.gif]]
| |
| − | | |
| − | Let the squares be labeled <math>A</math>, <math>B</math>, <math>C</math>, and <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>.
| |
| − | | |
| − | So, any "new" square that is attached to those edges will prevent the polygon from becoming a cube with one face missing.
| |
| − | | |
| − | Therefore, squares <math>1</math>, <math>2</math>, and <math>3</math> will prevent the polygon from becoming a cube with one face missing.
| |
| − | | |
| − | 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>.
| |
| − | | |
| − | 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>
| |
| − | | |
| − | == See Also ==
| |
| − | {{AMC10 box|year=2003|ab=A|num-b=9|num-a=11}}
| |
| − | | |
| − | [[Category:Introductory Geometry Problems]]
| |
