map $r_2$. We can think of $r_2$ as sending the rook in $P'$ to $P$ (arrow 1), sending the rook in $S$ to a place in $T$ via $s$ (arrow 2), and then projecting rooks in conflicting rows and columns to rows and columns that are known to be unoccupied (arrows 3 and 4).

\(P) at (0,-1); (P’) at (2,0); (S) at (0,-.5); (S’) at (1,0); (V) at (-1,-1); (V’) at (-1,-.5); (H) at (1,.5); (H’) at (2,.5);

(-.1,-1.3) node [$P$]{}; (P’) node [$P'$]{}; (-2,-.1) node [$i'$]{}; (-2,-1.1) node [$i$]{};

(1.9,-.2) to (0,-1.15); (1,-.75) node [$1$]{}; (0,-.35) to (.7,-.2); (.35, -.4) node [$2$]{}; (-1.3,-1) to (-1.3,-.4); (-1.4,-.7) node[$3$]{}; (.6,.6) rectangle (.8,.8); (.8,.7) to (1.8,.7); (1.3,.8) node [$4$]{};

(-.2,-1) grid (0,-.8); (-.1,-.7) – (-.1,-.5); (-.2,-.4) rectangle (0,-.2); (-.2,-.4) grid (0,0); (-.2,0) grid (0,.4); (-.1,.5) – (-.1,.7); (-.2,.8) grid (0,1); (-.2,.8) – (0,.8);

(1.8,0) grid (2,.2); (1.9,.3) –(1.9,.5); (1.8,.6) rectangle (2,.8); (1.8,.6) grid (2,1); (1.8,.6) – (1.8,1); (1.8,.6) – (2,.6); (1.8,0) – (1.8,.2);

(-2,-.2) grid (-1.2,0); (-1.1,-.1) – (-.9,-.1); (-.8,-.2) grid (0,0); (.6,-.2) rectangle (.8,0); (0,-.2) grid (.8,0); (1,-.1) – (1.2,-.1); (1.8,-.2) rectangle (2,0); (1.4,-.2) grid (2,0); (1.4,-.2) – (1.4,0); (-1.4,-.4) rectangle (-1.2 ,-.2); (-1.4,-.4) rectangle (-1.2 ,-.2);

(-1.4,-1.2) rectangle (-1.2,-1); (-2,-1.2) 