To solve the problem of finding the minimum number of colors needed to paint the cube's vertices such that adjacent vertices (sharing an edge) have different colors, we can analyze the cube as a graph:

Key Observations:

The cube's vertex graph is bipartite. A bipartite graph can be divided into two disjoint sets of vertices where no two vertices within the same set are adjacent.

For a cube, we can split the vertices into two groups based on the parity of the sum of their coordinates (e.g., $(x,y,z)$: sum even vs sum odd). Adjacent vertices differ in exactly one coordinate, so their sum parity flips—meaning no two adjacent vertices are in the same group.

Conclusion:

Since the cube is bipartite, it can be colored with 2 colors (one for each group), and we cannot use fewer than 2 colors (as there are edges between vertices).

Assuming the problem intended the minimum number of colors (a common question), the answer is:

2

$\boxed{2}$

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