jmc

Question

The six faces of a three-inch wooden cube are each painted red. The cube is then cut into one-inch cubes along the lines shown in the diagram. How many of the one-inch cubes have red paint on at least two faces? FIGURE

Accepted Answer

Only the 8 corners of the cube have three faces painted red. Each edge has one cube that has 2 faces painted red. There are 12 edges, so 12 cubes have 2 faces painted red. Each of the six faces has only the center cube painted on exactly 1 face, and the single cube in the center of the three-inch cube is the only one with no faces painted. We can thus produce the following table: array|c|c| Number of red faces & Number of one-inch cubes 3 & 8 2 & 12 1 & 6 0 & 1 2|r| Total = 27 array Number of cubes with 2 or 3 red faces is 8 + 12 = 20.