Twenty-seven identical white cubes are stacked together to form one larger cube. All the outside faces of ....

Question

Twenty-seven identical white cubes are stacked together to form one larger cube. All the outside faces of ....the larger cube are then painted red. If the larger cube is takes apart, then number of smaller cubes with exactly four white faces is...


Please explain your answer

Answer 1

The big cube is like a Rubik's Cube, a cube composed of 3x3x3 small ones.

Smaller cubes with exactly four white faces, are cubes that when assembled have two outward-facing (then painted red) faces. There are 12 small cubes sitting in "edge" positions in the big cube (4 around the top layer, 4 around the bottom layer, and 4 around the center) -- one for each edge of the big cube.

There are also six "center" pieces in the middle of a 3x3 face of the big cube that will end up with one red face, and eight "corner" pieces that will end up with three red faces. Finally there is one cube in the very center of the big cube that will have no red faces. All together: 12+8+6+1, accounts for all 27 small cubes.

Answer 2

My suggestion would be to draw a diagram to make this simpler, but let's see how it goes without one.

If you're going to paint all the outside faces, the cubes with two faces painted are the ones with four white faces. Think about that visually. Each edge piece that isn't a corner piece would have two red faces (corner pieces would have three, so rule those out). If it's hard to think of all those side pieces, think of everything else and see what you have left.

You have 27 to start with. The very center piece is obviously ruled out as it will be all while. You have 26. Then you can rule out the center piece of each face (they would have one red). That leaves you with 20. Now rule out the corners. There are eight of them, so you're left with twelve, which is the answer.

I'd advise drawing a picture and counting those twelve edges instead of doing what I did though, as this seems like an SAT question that you'd want to get through as fast as possible.

Answer 3

On each edge, you have one smaller cube with exactly four white faces. There are 12 edges in a cube. Therefore, there are 12 smaller cubes with exactly four white faces.

Answer 4

never mind

edge pieces not on the corner fit