Pentomino to Cube?
Source:
February 19, 2008
geometry3D geometry
Problem Statement
The polygon enclosed by the solid lines in the figure consists of 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("",(1.5,0.5));
draw(circle((1.5,0.5),.17));
label("",(2.5,1.5));
draw(circle((2.5,1.5),.17));
label("",(3.5,1.5));
draw(circle((3.5,1.5),.17));
label("",(4.5,2.5));
draw(circle((4.5,2.5),.17));
label("",(3.5,3.5));
draw(circle((3.5,3.5),.17));
label("",(2.5,3.5));
draw(circle((2.5,3.5),.17));
label("",(1.5,3.5));
draw(circle((1.5,3.5),.17));
label("",(0.5,2.5));
draw(circle((0.5,2.5),.17));
label("",(0.5,1.5));
draw(circle((0.5,1.5),.17));[/asy]