MATHCOUNTS Exercises – 9

Solution There are only 2 ways to paint 3 faces in blue color, as shown below with unfold cube faces:

For the first case where 3 blue faces form a strip, there are 3 ways to paint the remaining 3 faces, by painting the face opposite to the middle blue face in red, green, or yellow, as shown below, with the center blue face not displayed.

For the second case where 3 blue faces jointing at a corner, there are 2 ways to paint the remaining 3 faces, either in clockwise order or in counter-clockwise order, as shown below with the 3 blue faces hiding in the back.

Therefore, there are 5 unique ways to paint the cube, as show below with unfold faces: