2021 AMC 10 B Fall
Complete problem set with solutions and individual problem pages
A cube is constructed from white unit cubes and 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.)(2021 AMC Fall 10B, Question #24)
- A.
- B.
- C.
- D.
- E.
Solution 1:
This problem is about the relationships between the white unit cubes and the blue unit cubes, which can be solved by Graph Theory. We use a Planar Graph to represent the larger cube. Each vertex of the planar graph represents a unit cube. Each edge of the planar graph represents a shared face between neighboring unit cubes. Each face of the planar graph represents a face of the larger cube. Now the problem becomes a Graph Coloring problem of how many ways to assign vertices blue and vertices white with Topological Equivalence. For example, in Figure (1), as long as the blue vertices belong to the same planar graph face, the different planar graphs are considered to be topological equivalent by rotating the larger cube.
Here is how the 4 blue unit cubes are arranged: In Figure (1): 4 blue unit cubes are on the same layer (horizontal or vertical). In Figure (2): 4 blue unit cubes are in shape. In Figure (3) and (4): 4 blue unit cubes are in shape. In Figure (5): 3 blue unit cubes are in shape, and the other is isolated without a shared face.
In Figure (6): 2 pairs of neighboring blue unit cubes are isolated from each other without a shared face. In Figure ( 7 blue unit cubes are isolated from each other without a shared face. So the answer is (A) 7
Solution 2:
Let's split the cube into two layers; a bottom and top. Note that there must be four of each color, so however many number of one color are in the bottom, there will be four minus that number of the color on the top. We do casework on the color distribution of the bottom layer.
Case 1: In this case, there is only one possibility for the top layer - all of the other color - . Therefore there is 1 construction from this case.
Case 2: In this case, the top layer has four possibilities, because there are four different ways to arrange it so that it also has a 3, 1 color distribution - . Therefore there are 4 constructions from this case.
Case 3: In this case, the top layer has six possibilities of arrangement - . However, having adjacent colors one way can be rotated to having adjacent colors any other way, so there is only one construction for the adjacent colors subcase and similarly, only one for the diagonal color subcase. Therefore the total number of constructions for this case is . The total number of constructions for the cube is thus
