# The game she loves me she loves me knot is played

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3. The gameShe-loves-me, she-loves-me-knotis played with a number of pieces of string. A move is to tie twoloose ends of string together. The game ends when all the ends are tied together, forming one or more loops.The winner is the player who ties the last knot. For instance, a game beginning with 4 pieces of string mightend like this:Find the sequence of Sprague-Grundy values for this game, beginning withnpieces of string, wherenis anonnegative integer. You donotneed to give a justification using induction, but must explain your solution.If we begin with 97 pieces of string, who wins the game (is this aPposition or anNposition)?(6 points)
4Name:(9 points)4.(a) Consider the 2×3 game of Chomp whose graph of possible positions is shown below. Indicate on the graphthe Sprague-Grundy values of all the positions, and mark whether the position is in N or in P. (Recallthat in Chomp, 2 players take turns biting off chunks of a rectangular chocolate bar that is divided intosquares (indicated here by circles). Each player, on their turn, removes an uneaten square along with allthe squares that lie above and to the right of it. The bottom left square, colored gray, is poisoned, so theplayer to chomp that one loses.)(4 points)(b) Now consider two people playing a Chomp game using 4 bars of chocolate. On a turn, a player picksone