Exercise 28

The setting for this problem is the fictional town of Mascotville, which is laid out as a grid. Mascots are allowed to travel only on the streets, and not “as the yellow jacket flies.” Buzz, the Georgia Tech mascot, wants to go visit his friend Thundar, the North Dakota State University mascot, who lives $6$ blocks east and $7$ blocks north of Buzz's hive. However, Uga VIII has recently moved into the doghouse $2$ blocks east and $3$ blocks north of Buzz's hive and already has a restraining order against Buzz. There's also a pair of tigers (mother and cub) from Clemson who live $1$ block east and $2$ blocks north of Uga VIII, and they're known for setting traps for Buzz. Buzz wants to travel from his hive to Thundar's pen every day without encountering Uga VIII or The Tiger and The Tiger Cub. However, he wants to avoid the boredom caused by using a route he's used in the past. What is the largest number of consecutive days on which Buzz can make the trip to visit Thundar without reusing a route (you may assume the routes taken by Buzz only go east and north)?