A painted cube cut into 27

Question

A solid wooden cube is painted red on all six faces, then cut into 27 equal small cubes (3 by 3 by 3).

How many of the small cubes have paint on exactly two faces? Also count those with paint on three faces, exactly one face, and no faces, and check that your four numbers add up to 27.

Accepted Answer

12 small cubes have exactly two painted faces. The full count: 8 with three faces, 12 with two, 6 with one, 1 with none. By position on the big cube. - Three painted faces: the small cubes at the corners. A cube has 8 corners. 8. - Two painted faces: the small cubes in the middle of each edge, not at a corner. A cube has 12 edges, and on a 3 by 3 by 3 cube each edge has exactly one middle cube. 12. - One painted face: the small cube at the centre of each face. Six faces. 6. - No painted faces: the single cube hidden in the middle. 1. Check: 8 + 12 + 6 + 1 = 27. For a bigger cube. If the cube …

A painted cube cut into 27

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