So if we impose a cartesian coordinate system with the unit square at (0,0) (0,1) (1,1) and (1,0). Then a point (x,y) has to meet the constraint |x|+|y|+|x-1|+|y-1|=4 Note that there are nine regions we have to consider; 8 are outside and around the square and 1 inside the square. Of the 8 outside regions, there are 4 of each of 2 types: the corner and edge regions. Inside the square the LHS is always 2. Consider the northern edge region, in that region |x|+|x-1|=1 and |y|+|y-1|=2y-1. 2y-1=3, $\boxed{y=2}$. Consider the northeastern corner region. In that region |x|+|x-1|=2x-1 and likewise for y. We have 2x+2y=6; $\boxed{x+y=3}$. You can rotate these two equations around (.5,.5) appropriately to get that the loci of points is the edge of an octagon with sidelengths of $1$ and $\sqrt{2}$