2020 AMC 12B Problems/Problem 20
Two different cubes of the same size are to be painted, with the color of each face being chosen independently and at random to be either black or white. What is the probability that after they are painted, the cubes can be rotated to be identical in appearance?
Define two ways of painting to be in the same if one can be rotated to form the other.
We can count the number of ways of painting for each specific .
Case 1: Red-blue color distribution is 0-6 (out of 6 total faces)
Trivially way to paint the cubes.
Case 2: Red-blue color distribution is 1-5
Trivially all ways belong to the same , so ways to paint the cubes.
Case 3: Red-blue color distribution is 2-4
There are two for this case: the where the two red faces are touching and the other where the two red faces are on opposite faces. There are members of the latter since there are unordered pairs of opposite faces of a cube. Thus, there are members of the former . Thus, ways to paint the cubes for this case.
Case 4: Red-blue color distribution is 3-3
By simple intuition, there are also two for this case, the where the three red faces meet at a single vertex, and the other class where the three red faces are in a "straight line" along the edges of the cube. Note that since there are vertices in a cube, there are members of the former class and members of the latter class. Thus, ways to paint the cubes for this case.
Note that by symmetry (since we are only switching the colors), the number of ways to paint the cubes for Red-blue color distributions 4-2, 5-1, and 6-0 is 2-4, 1-5, and 0-6 (respectively).
Thus, our total answer is
(The above solution explains the solution very well; however the thought process is not explicitly addressed)
We want P(two cubes can be rotated to match after they are painted); i.e. P(cubes are rotationally identical). By Polya's enumeration theorem (from Burnside's Lemma), there are distinct ways to color a cube using colors. Plugging in , we find that there are 10 distinct ways to color a cube with 2 colors (meaning none of these 10 colorations can be rotated to match another). However, each coloration does not have the same probability of occuring (the cubes are painted randomly); for example, there is only 1 way to paint both cubes all black, as any rotations will not yield a visually different result. There are multiple ways to paint each cube with 1 white face and 5 black faces, as one cube can simply be rotated to match the other. Therefore, we have to do casework for 10 cases, where each case is a distinct way to color a cube. (The above solution sometimes addresses more than one case at once)
Case 1: Both cubes are completely black 1 way, P(Case 1) = (There are 2^12 or 4096 total colorations of the cubes)
Case 2: Exactly one face is white, the rest are black 6 ways, P(Case 2) =
Case 3: 2 white faces share an edge, the rest are black = 12 ways, P(Case 3) =
Case 4: 2 white faces opposite each other, the rest are black = 3 ways, P(Case 4) =
Case 5: 3 white faces share a vertex, the rest are black = 12 ways, P(Case 5) =
Case 6: 3 white faces in a "row", the rest are black = 8 ways, P(Case 5) =
Cases 7, 8, 9, and 10: These cases are the same as cases 1, 2, 3, and 4, albeit with a color swap. Doubling the probabilities of cases 1-4 will cover these cases.
Therefore we have P(cubes are rotationally identical) = =
P.S. Sorry for the bad formatting. My latex isn't very good =(
Video Solution by TheBeautyofMath
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