Exercise 1.4. The impartial game Cram is played on a board of m x n squares, where players alternately place a domino on the board which covers two adjacent squares that are free (not yet occupied by a domino), vertically or horizontally. The first player who cannot place a domino any more loses. Example play for a 2 x3 board: I I II loses (a) Who will win in 3 x 3 Cram? [Hint: Use the symmetry of the game to investigate possible moves, and remember that it suffices to find one winning strategy.) (b) Who will win in m x n Cram when both m and n are even? (That player has a simple and precisely definable winning strategy which you have to find.) (c) Who will win in m x n Cram when m is odd and n is even? (Easy with (b).) Note (not a question): Because of the known answers from (b) and (c), this game is more interesting for real play on an m x n board where both m and n are odd. Play it with your friends on a 5 x 5 board, for example. The situation often decomposes into independent parts, like contiguous fields of 2, 3, 4, 5, 6 squares, that have a known winner, which may help you analyse the situation. Exercise 1.7. Consider the game Cram from Exercise 1.4, played on a 1 x n board for n > 2. Let Dn be the Nim value of that game, so that the starting position of the 1 x n board is equivalent to a Nim heap of size Dn. For example, D2 = 1 because the 1 x 2 board is equivalent to *1. (a) How is Dn computed from smaller values Dk for k 2. Let Dn be the Nim value of that game, so that the starting position of the 1 x n board is equivalent to a Nim heap of size Dn. For example, D2 = 1 because the 1 x 2 board is equivalent to *1. (a) How is Dn computed from smaller values Dk for k