Write $N$ for $2022$. We make a graph with the numbers $1, 2, \dots, N$ as vertices. Whenever two black balls numbered $a, b$ are put in the same black box, we add a black edge between the vertices $a$ and $b$; similarly add white edges corresponding to white boxes. In the resulting graph, each vertex has two edges attached to it. Thus the graph consists of several circles. Moreover, since each vertex has exactly one black edge and one white edge, the edges in each circle must have alternating color, and consequently the length of the circle must be even. Now we want to paint each vertex black or white, in such a way that any two vertices connected by an edge have different colors. This is obviously possible, given the above condition on the graph.