smc

Let the squares be labeled $A$, $B$, $C$, and $D$. When the polygon is folded, the "right" edge of square $A$ becomes adjacent to the "bottom edge" of square $C$, and the "bottom" edge of square $A$ becomes adjacent to the "bottom" edge of square $D$. So, any "new" square that is attatched to those edges will prevent the polygon from becoming a cube with one face missing. Therefore, squares $1$, $2$, and $3$ will prevent the polygon from becoming a cube with one face missing. Squares $4$, $5$, $6$, $7$, $8$, and $9$ will allow the polygon to become a cube with one face missing when folded. Thus the answer is $\boxed{6}$.