How many small cubes are there with at least two different colors on their faces?
Explanation:
All the edge (including corner) cubes will give us cubes with 2 faces painted.
Total edge cubes = 12n – 2 × 8 = 60 – 16 = 44.
Now, there will be 9 cubes, 3 each along the 3 edges which have same color both sides, which will have only 1 color on them.
∴ Number of cubes with two different colors on their faces = 44 – 9 = 35.
Alternately,
There are 9 edges with different colors on both sides each with 3 middle pieces.
∴ These 9 edges will give 9 × 3 = 27 cubes with 2 different colors on them.
There are also 8 corners all with 2 different colors on them.
∴ Number of cubes with two different colors on their faces = 27 + 8 = 35.
Hence, 35.
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