# Two rooks for Bobby Fischer

Bobby Fischer liked to play the following game on a standard $8\times8$ chessboard:

• In his first step, Bobby placed a white rook and a black rook somewhere on the chessboard (on two different squares, of course).
• In every further step, Bobby picked one of the rooks, and then moved it away from its current square to a horizontally or vertically adjacent (currently empty) square.

Is it possible that after $64\cdot63=4032$ steps, one has generated each of the $4032$ possible positions for the two rooks exactly once?

Call a position even if the two rooks are on squares of the same color squares, odd if they are on squares of different colors. Every move changes the position from even to odd or from odd to even. There are a total of $31\cdot 64=1984$ even positions and $32\cdot 64=2048$ odd positions, so it is not possible to reach them all while alternating between them.