Problem #2099

 2099 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? $[asy] unitsize(10mm); defaultpen(fontsize(10pt)); pen finedashed=linetype("4 4"); filldraw((1,1)--(2,1)--(2,2)--(4,2)--(4,3)--(1,3)--cycle,grey,black+linewidth(.8pt)); draw((0,1)--(0,3)--(1,3)--(1,4)--(4,4)--(4,3)-- (5,3)--(5,2)--(4,2)--(4,1)--(2,1)--(2,0)--(1,0)--(1,1)--cycle,finedashed); draw((0,2)--(2,2)--(2,4),finedashed); draw((3,1)--(3,4),finedashed); label("1",(1.5,0.5)); draw(circle((1.5,0.5),.17)); label("2",(2.5,1.5)); draw(circle((2.5,1.5),.17)); label("3",(3.5,1.5)); draw(circle((3.5,1.5),.17)); label("4",(4.5,2.5)); draw(circle((4.5,2.5),.17)); label("5",(3.5,3.5)); draw(circle((3.5,3.5),.17)); label("6",(2.5,3.5)); draw(circle((2.5,3.5),.17)); label("7",(1.5,3.5)); draw(circle((1.5,3.5),.17)); label("8",(0.5,2.5)); draw(circle((0.5,2.5),.17)); label("9",(0.5,1.5)); draw(circle((0.5,1.5),.17));[/asy]$ $\mathrm{(A) \ } 2\qquad \mathrm{(B) \ } 3\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 5\qquad \mathrm{(E) \ } 6$ This problem is copyrighted by the American Mathematics Competitions.
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• Reduce fractions to lowest terms and enter in the form 7/9.
• Numbers involving pi should be written as 7pi or 7pi/3 as appropriate.
• Square roots should be written as sqrt(3), 5sqrt(5), sqrt(3)/2, or 7sqrt(2)/3 as appropriate.
• Exponents should be entered in the form 10^10.
• If the problem is multiple choice, enter the appropriate (capital) letter.
• Enter points with parentheses, like so: (4,5)
• Complex numbers should be entered in rectangular form unless otherwise specified, like so: 3+4i. If there is no real component, enter only the imaginary component (i.e. 2i, NOT 0+2i).