SOLUTION: below is a 3by3by3cube not all of the cubes are visible. suppose the entire outside of the cube is painted red including the bottom. how many different 2 by 2 by 2 cubes with exact

Question 552409: below is a 3by3by3cube not all of the cubes are visible. suppose the entire outside of the cube is painted red including the bottom. how many different 2 by 2 by 2 cubes with exactly three red faces can be found in the shown cube?
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
Your picture did not show. It must have looked something like this, but with 9 squares on each face of the cube
And my picture should look like one of the 2 by 2 by 2 cubes found in the large cube.
A given face of a cube has 4 neighbor faces, sharing an edge. Only the 6th face has no common edge; it is opposite and parallel to the first (given) face. So, in any cube with exactly 3 red faces, at least two of the red faces must share an edge.
In 2 by 2 by 2 cubes cut from the painted 3 by 3 by 3 cube, the 3 red faces must share are vertex, because we cannot have two red faces parallel to each other. So the 2 by 2 by 2 cubes must include a corner/vertex of the original painted 3 by 3 by 3 cube. Since there are only 8 such corners, there will be only 8 different 2 by 2 by 2 cubes with exactly 3 red faces.
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