A "By Hand" Example

This small example provides a convenient comparison to the computer algorithm adaptation later. Consider the following chessboard of four cells and (arbitrarily) make the special cell at the start. The choice of intermediate special cells is obvious. The following reduction leads to the unique rook polynomial, .

The steps taken use algebraic combination and the knowledge that the rook polynomials of the empty chessboard and a single cell (in this case ) are 1 and .