2003 AMC 10A Problem 10

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: 1410

Small hint:

The four squares already wrap into four side faces of a cube

Big hint:

The fifth square must fold up as one of the two remaining faces without landing on a face already covered

Solution:

Fold the four shaded squares first. They occupy four distinct faces of the cube, leaving two faces open.

Checking the numbered attachments, positions

1,

2,

and

3

fold the new square onto a face already occupied by one of the shaded squares. Positions

4,

5,

6,

7,

8,

and

9

fold it onto one of the two open faces. Therefore exactly

6

of the

9

polygons work.

Thus, the correct answer is E.