Covering the surface of a 3 x 3 x 3 cube with 3 x 1 strips of paper, what is the greatest possible number of bent strips? There is a wooden $3 \times 3 \times 3$ cube and 18 rectangular $3 \times 1$ paper strips. Each strip has two dotted lines dividing it into three unit squares. The full surface of the cube is covered with the given strips, flat or bent. Each flat strip is on one face of the cube. Each bent strip (bent at one of its dotted lines) is on two adjacent faces of the cube. What is the greatest possible number of bent strips? Justify your answer.
Attemp: 14!
14+ doesnt work because each corner needs a flat strp and there are 8 corners and each strap can cover 2 corners. I think I proved that the number of bent strips has to be even and no greater than 14.
 A: The cube has 12 edges, each split into 3 segments, so 36 edge segments in total. Every bent strip covers one of them. As you already noted, at each corner of the cube at least one of the adjacent facelets must be covered by a flat strip, so we need (at least) 4 flat strips running from corner to corner. Note that such a flat strip between two corners will block 5 edge segments, and however you cover the rest of the two corners with two bent strips, at least one further edge segment becomes blocked (*). So it would seem that the four flat strips cause at least 24 edge segments to be blocked, leaving at most 36-24=12 to be used for the bent strips. That would make 12 bent strips the maximum.
An arrangement with 12 bent strips is fairly easy, for example cover the top/bottom faces with straight strips, and cover the sides of each layer by bent strips all around.
However, there is a flaw in the above proof. That sixth edge segment which is blocked by one of the bent strips (see * above), is blocked by the facelet on one side being filled. We can let the other side be blocked in the same way, and have therefore double-counted them. This leads to the following solution:
                k k l
                n o o
                b c c
  J J J  k n b  A A A  c o l  
  i m m  m n b  d q p  p p l  
  i r f  E E E  d q g  H H H  
         f f d
         r r q
         i g g

The four flat strips have capital letters (A, E, H, J), and the other 14 are bent.
