The Duke of Squares left to his three sons a square estate, $100\times 100$ square miles, made up of ten thousand $1\times 1$ square mile square plots. The whole estate was divided among his sons as follows. Each son was assigned a point inside the estate. A $1\times 1$ square plot was bequeathed to the son whose assigned point was closest to the center of this square plot. Is it true that, irrespective of the choice of assigned points, each of the regions bequeathed to the sons is connected (that is, there is a path between every two of its points, never leaving the region)?