AMC10 2021 10B, Problem 24

A cube is constructed from $4$ white unit cubes and $4$ blue unit cubes. How many different ways are there to construct the $2 \times 2 \times 2$ cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.)

Solution: 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 $2$ 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 $4$ vertices blue and $4$ vertices white with Topological Equivalence. For example, in Figure $(1)$, as long as the $4$ 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 $T$ shape. In Figure $(3)$ and $(4)$: $4$ blue unit cubes are in $S$ shape. In Figure $(5)$: $3$ blue unit cubes are in $L$ 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 $(7)$: $4$ blue unit cubes are isolated from each other without a shared face. So the answer is $\fbox{7}$

Answer: 7