2003 AMC 10A Problems/Problem 10
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 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?
Solution
Let the squares be labeled , , , and .
When the polygon is folded, the "right" edge of square becomes adjacent to the "bottom edge" of square , and the "bottom" edge of square becomes adjacent to the "bottom" edge of square .
So, any "new" square that is attached to those edges will prevent the polygon from becoming a cube with one face missing.
Therefore, squares , , and will prevent the polygon from becoming a cube with one face missing.
Squares , , , , , and will allow the polygon to become a cube with one face missing when folded.
Thus the answer is .
Another way to think of it is that a cube missing one edge has 5 of it's 6 faces. Since the shape has 4 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. 1,2, and 3 overlap, while 4 9 do not. The answer is 6
See Also
2003 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
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All AMC 10 Problems and Solutions |