VII. (12 points) What is the proper first move in a Nim game with piles of 3, 5, and 8 chips? Explain your answer. VIII. (12 points) Which player (Player 1 or Player 2) should win the Nim game with piles of 2, 4, and 6 chips? Explain your answer. IX. (11 points) If Nim piles of sizes 11, 14, and x have a Nim sum of 0, then what is the value of x? Explain your answer. BONUS (5 points extra credit): Eight nodes of a network are placed in a horizontal line, as follows: 0 0 0 O 0 0 0 0 Two players alternate drawing an undirected arc between two consecutive nodes. The first player to form a path on three nodes loses. Show that Player 1 has a guar- anteed winning strategy for this game. What changes if there are 10 nodes? O O 0 0 0 0 0 0 0 0