It started as a fairly simple proposition: There were two schools on the Upper West Side of Manhattan, one overcrowded, the other underused. The city’s Education Department proposed to redraw the schools’ attendance zone so that some blocks assigned to the crowded school would be shifted to the emptier one.

But while most of the children at the crowded school, Public School 199, are middle class and white, most of the children at the other school, P.S. 191, are poor, and black or Hispanic. P.S. 191 has much lower test scores, and last year, the state labeled the school persistently dangerous, though many of its supporters argued that this was a mistake.

The proposal became complicated as families in the blocks that would be affected objected to the change. The department ultimately dropped the idea.

Now, instead of a simple solution, the department is considering a convoluted one that amounts to an educational game of musical chairs: First, P.S. 191 would move a block west, taking over a building under construction that was originally intended for a new school. The hope is that the move would provide a symbolic fresh start for P.S. 191 and that the gleaming campus would make it more appealing to the families moved there. Then, another school on the Upper West Side, P.S. 452, which shares a building with two other schools, would move into P.S. 191’s current home, giving it room to grow. The school that had been envisioned for the new building would not open.