7. You are an intern at Stark Industries working at Avengers Tower. As an eccentric man, Tony Stark designed his building in a peculiar shape where the n-th floor consists of n rooms arranged into an n x n array. For example, the floor plans of the third and fourth floors are illustrated below. Here you can only travel back and forth between every two rooms if they are adjacent, i.e. sharing a wall F3 F4 One day, there is an intruder inside the tower so you and your fellow intern Peter Parker have to lock down every room to prevent the Avengers' technologies from falling into the wrong hands. Once a room is locked with Peter's special ability, it is no longer available and you cannot revisit it. Furthermore, Peter can only lock down a room from the inside, so in order to fully lock down a floor, he has to visit every room on that floor. Suppose that you start at a room adjacent to a corner room. Prove that when n 2 3 and n is odd, then it is impossible to lock down every room on the n-th floor in this case. When n 2 3 and n is even, is it possible to lock down every room on the n-th floor if you start at a room adjacent to a corner room? Illustrate on the map of floors 4,6, and 8 "Mr. Stark, I don't feel so good..." - Peter's last words before getting fired from the internship 7. You are an intern at Stark Industries working at Avengers Tower. As an eccentric man, Tony Stark designed his building in a peculiar shape where the n-th floor consists of n rooms arranged into an n x n array. For example, the floor plans of the third and fourth floors are illustrated below. Here you can only travel back and forth between every two rooms if they are adjacent, i.e. sharing a wall F3 F4 One day, there is an intruder inside the tower so you and your fellow intern Peter Parker have to lock down every room to prevent the Avengers' technologies from falling into the wrong hands. Once a room is locked with Peter's special ability, it is no longer available and you cannot revisit it. Furthermore, Peter can only lock down a room from the inside, so in order to fully lock down a floor, he has to visit every room on that floor. Suppose that you start at a room adjacent to a corner room. Prove that when n 2 3 and n is odd, then it is impossible to lock down every room on the n-th floor in this case. When n 2 3 and n is even, is it possible to lock down every room on the n-th floor if you start at a room adjacent to a corner room? Illustrate on the map of floors 4,6, and 8 "Mr. Stark, I don't feel so good..." - Peter's last words before getting fired from the internship