A square made of cardboard has sides of unit length and corners marked $A , B , C , D$.\ (a) Show that there are eight different ways of placing the cardboard square so that it completely covers the square region in the $( x , y )$ plane with corners at the points $( 0,0 ) , ( 0,1 ) , ( 1,1 ) , ( 1,0 )$.\ (b) Initially the square is positioned so that $A , B , C , D$ are over the points $( 0,0 ) , ( 0,1 )$, $( 1,1 ) , ( 1,0 )$, respectively. You may move the square by either\ (i) rotating it in the plane by $90 ^ { \circ }$ about one of the corners, or\ (ii) turning it over keeping one of the edges fixed in contact with the plane. Show that after two moves it is possible to return the square to its initial position but with the corners $B$ and $D$ interchanged.\ (c) Show that four of the eight configurations in (a) can be achieved from the initial position and moves described in (b).
A square made of cardboard has sides of unit length and corners marked $A , B , C , D$.\
(a) Show that there are eight different ways of placing the cardboard square so that it completely covers the square region in the $( x , y )$ plane with corners at the points $( 0,0 ) , ( 0,1 ) , ( 1,1 ) , ( 1,0 )$.\
(b) Initially the square is positioned so that $A , B , C , D$ are over the points $( 0,0 ) , ( 0,1 )$, $( 1,1 ) , ( 1,0 )$, respectively. You may move the square by either\
(i) rotating it in the plane by $90 ^ { \circ }$ about one of the corners, or\
(ii) turning it over keeping one of the edges fixed in contact with the plane.
Show that after two moves it is possible to return the square to its initial position but with the corners $B$ and $D$ interchanged.\
(c) Show that four of the eight configurations in (a) can be achieved from the initial position and moves described in (b).