The painted cube cut into 64 cubies

Question

A solid wooden cube is painted red on all six outer faces. Then it is sliced into a 4 by 4 by 4 grid, producing 64 identical small cubes ("cubies").

How many cubies have paint on exactly 3 faces, exactly 2 faces, exactly 1 face, and no faces at all?

Accepted Answer

Classify the cubies by where they sat in the big cube. Paint only reached the outer surfaces, so a cubie's painted-face count equals how many outer faces of the big cube it touched. 3 painted faces, the corners. Only the 8 corner cubies touch three outer faces. That's 8 cubies. 2 painted faces, the edges. Cubies along an edge (but not the corners) touch two faces. Each of the 12 edges of a cube, with the 2 corners removed, has 4 - 2 = 2 such cubies. So 12 	imes 2 = \mathbf{24} cubies. 1 painted face, the face centers. Cubies in the middle of each face touch one outer face. Each face has an i…

The painted cube cut into 64 cubies

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