2003 AMC 12A Problem 13

Problem 13 of 25IntermediateGeometry

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?

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Solution

A cube missing one face has

5

faces, so the fifth square must fold onto a face not already occupied by the four-square piece. Tracing the face reached by each attachment, positions

1,

2,

and

3

fold onto a face already occupied and overlap it. Each of positions

4,

5,

6,

7,

8,

and

9

folds onto an unoccupied face. Therefore

6

of the nine positions work. Thus, the correct answer is E.

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