The Generalization of Rook Number r 2 for the Fractal Chessboard

In this paper we develop a generalized formula of r2, the number of ways of placing two non-attacking Rooks for the Fractal Chessboard which is defined as a board that grows progressively in a consistent manner using a 2 × 2 chessboard to its sides and corners. The board is disintegrated into small sub boards based on their position in the whole Fractal Chessboard (FC). The board is disintegrated into sub boards based on their position in the whole board FC. By finding the value of r2 for each of these sub boards and adding them, the r2 value of the whole board FC is obtained. Finally the r2 value is generalized the Fractal Chessboard at any iteration I ≥ 4 .