Consider the game of up-and-down rooks played on a standard chess- board. player i has a set of white rooks initially located in the bottom row, while player ii has a set of black rooks in the top row. in each turn, a player moves one of its rooks up or down a column, without skipping over the other rook or occupying its position. the first player who cannot move loses. this game is not progressively bounded, yet an optimal strategy exists. find such a strategy by relating this game to a nim position with 8 piles.