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?

$2$

$3$

$4$

$5$

$6$

Solution:

When the four given squares are folded, two pairs of their edges meet to form a band of four faces around the cube, leaving two faces open.

The fifth square folds into one of these open faces exactly when it is attached along a free edge. This works for $6$ of the $9$ positions; the other $3$ would fold onto a face already covered.

Thus, the correct answer is E.

2003 AMC 10A Problem 10

Solution:

Problem 10 in Other Years