Four congruent squares form the gray T-shaped figure shown. A fifth congruent square is attached at one of the eight labeled positions. How many of the resulting shapes cannot be folded to form a cube with one face missing?
Take the middle square of the row of three as the bottom of the box. The squares to its left and right become the left and right walls, and the square below it becomes the front wall. The fifth square must become the back or the top.
Positions $1,2,3$ (above the row) fold to the back face; positions $4$ and $5$ (at the ends of the row) fold to the top, attached to the top edges of the left and right walls; position $8$ (below the stem) folds to the top, attached to the top edge of the front wall. These $6$ positions work.
Positions $6$ and $7$ (below the left and right squares) are attached to the front edges of the side walls, so they fold onto the front face, which is already occupied. Thus $2$ of the eight placements cannot be folded into a cube with one face missing.
