Big Rubiks Cube
An
cube is painted red on 3 faces and blue on 3 faces such that no corner is surrounded by
three faces of the same color. The cube is then cut into 512 unit cubes. How many of these cubes
contain both red and blue paint on at least one of their faces?
The answer is 56.
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The solution in my mind for this is visual; as I can't draw diagrams on my phone, I'm not sure if I can convey it successfully, but I'll try:
Due to the symmetric nature of a cube, there's only one way to paint it that meets the terms of the question. A side, the opposite side and any one of the other sides. Of the twelve edges, two are where red sides meet, two are where blue sides meet and eight are where blue meets red.
Each edge has six edge cubes (plus two shared corner cubes), so there are 6x8=48 edge cubes with both colors. Add on the eight corner cubes for a solution of 56.