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PrintFall 2021 AMC 10 B
United States 2021 counting and probability
Problem
A cube is constructed from 4 white unit cubes and 4 blue unit cubes. How many different ways are there to construct the cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.) (A) 7 (B) 8 (C) 9 (D) 10 (E) 11
Solution
Rotate the cube so that the number of small blue cubes showing in the front face is maximized.
If the front face contains four blue cubes, this gives 1 possible construction.
If the front face contains three blue cubes, then the back face must contain one blue cube, which can be in any of 4 positions. None of these constructions can be rotated into any other, so this gives 4 possible constructions. Here is why the four placements of the one blue cube on the back face lead to different patterns. Assume without loss of generality that the sole white cube on the front face is in the upper right. There is a second face with three blue cubes if and only if the back cube is placed in any position on the back face except for the upper right. Hence if the blue cube on the back face is placed at the upper right, then there will be a blue cube adjacent to no other blue cubes, and this does not occur with any other placement of the blue cube on the back face. If the blue cube on the back face is placed at the lower left, then there is one blue cube with three blue neighbors, but not if it is placed anywhere else. This leaves two cases, namely when the blue cube on the back face is placed at the upper left or lower right. In either of these cases, if the cube is rotated so that the unique other face having three blue cubes is placed in the front with the sole white cube on that face in the upper right, then the same configuration of blue and white cubes is obtained, rather than the other one. This gives 4 possible constructions.
Otherwise, each face of the large cube contains exactly two blue cubes. If some face contains two blue cubes in adjacent positions in a row or column, then rotate the cube to make that face the front face. Then the back face must contain two blue cubes in the diagonally opposite row or column, respectively. This gives 1 possible construction.
Otherwise, no blue cubes are in adjacent positions, and every face of the large cube is colored blue and white in checkerboard fashion. There is just 1 way to do this.
Thus the total number of possible constructions is . The figure below shows the four possible cases.
If the front face contains four blue cubes, this gives 1 possible construction.
If the front face contains three blue cubes, then the back face must contain one blue cube, which can be in any of 4 positions. None of these constructions can be rotated into any other, so this gives 4 possible constructions. Here is why the four placements of the one blue cube on the back face lead to different patterns. Assume without loss of generality that the sole white cube on the front face is in the upper right. There is a second face with three blue cubes if and only if the back cube is placed in any position on the back face except for the upper right. Hence if the blue cube on the back face is placed at the upper right, then there will be a blue cube adjacent to no other blue cubes, and this does not occur with any other placement of the blue cube on the back face. If the blue cube on the back face is placed at the lower left, then there is one blue cube with three blue neighbors, but not if it is placed anywhere else. This leaves two cases, namely when the blue cube on the back face is placed at the upper left or lower right. In either of these cases, if the cube is rotated so that the unique other face having three blue cubes is placed in the front with the sole white cube on that face in the upper right, then the same configuration of blue and white cubes is obtained, rather than the other one. This gives 4 possible constructions.
Otherwise, each face of the large cube contains exactly two blue cubes. If some face contains two blue cubes in adjacent positions in a row or column, then rotate the cube to make that face the front face. Then the back face must contain two blue cubes in the diagonally opposite row or column, respectively. This gives 1 possible construction.
Otherwise, no blue cubes are in adjacent positions, and every face of the large cube is colored blue and white in checkerboard fashion. There is just 1 way to do this.
Thus the total number of possible constructions is . The figure below shows the four possible cases.
Final answer
A
Techniques
Enumeration with symmetryColoring schemes, extremal arguments