2003 AMC 12A Problem 13

Question

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?

2

3

4

5

6

Answer: E

Concepts:net (3D geometry)cube geometry

Difficulty rating: 1530

Small hint:

A cube with one face missing needs

5

squares

Big hint:

Fold the four-square piece partway around a cube, then check which attached square lands on a still-open face

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.