A certain solid cube n inches on a side is made of nice white pine and is painted yellow on the outside. The cube is then cut up into n3 unit cubes; these little fellows have 0, or 1, or 2, or 3 yellow faces. It happens that the number of unit cubes with 1 yellow face is exactly twice the number of those having 2 yellow faces, and there are exactly eight times as many all-white cubes as there are cubes with 3 yellow faces. Find n.
The cube is n x n x n.
8 cubes have 3 yellow faces (the vertices of the big cube).
12(n – 2) cubes have 2 yellow faces (the edges of the big cube).
6(n – 2)2 cubes have 1 yellow face (centers of the faces of the big cube).
(n – 2)3 cubes have 0 yellow faces (the cube inside the big one).
12(n – 2) cubes have 2 yellow faces (the edges of the big cube).
6(n – 2)2 cubes have 1 yellow face (centers of the faces of the big cube).
(n – 2)3 cubes have 0 yellow faces (the cube inside the big one).
Check: (n – 2)3 + 6(n – 2)2 + 12(n – 2) + 8 = n3
→ n3 – 6n2 + 12n – 8 + 6n2 – 24n + 24 + 12n – 24 + 8 = n3
So, we set 6(n – 2)2 = 2[12(n – 2)] → 6n2 – 24n + 24 = 24n – 48
→ 6n2 – 48n + 72 = 0 → n2 – 8n + 12 = 0
→ (n – 2)(n – 6) = 0
→ (n – 2)(n – 6) = 0
If n = 2, the first condition is met (0 = 2 x 0) but not the second (0 ≠ 8 x 8).
If n = 6, both conditions are met:
48 cubes have 2 yellow faces,
96 cubes have 1, and
64 have 0 yellow faces.
96 cubes have 1, and
64 have 0 yellow faces.