Combinatorics Intermediate Challenges-III
How many different ways can you paint the faces of a cube using no more than two specific colors, say, black and white?
This problem is not original, and is roughly the level of difficulty of a problem in the Intermediate Challenge.This problem is part of this set .
The answer is 10.
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1 solution
Since there are only two colors (say black and white), we can consider cases from painting 0-3 faces and just double it.
2^6 is not the right answer as there are many cases of repetition.
Eg: A case with 4 white faces will also be a case with 2 black faces.
Working with white:
0 Faces: 1 way
1 Face: 1 way (all faces are similar)
2 Faces: 2 ways (white on opposite sides, and on adjacent sides)
3 Faces: 2 ways Lets consider two faces are already painted. - If the faces are on opposite sides, there is only one way to paint the remaining side, as a bridge. - If they are adjacent, there are two ways to paint the new face such that it a. touches one of the other two faces (repetition of bridge case) b. touches both the other faces.
So total ways is
2 x (1 + 1 + 2) + 2 (number of ways to paint 3 faces is symmetric so not multiplied by two) = 2 x 4 + 2 = 10 ways.