Beyond Euclidean Geometry

Question

If you can please help me with three questions for geometry work. The link I posted contains the snaps of all three questions. Thank you so much!

http://s1039.photobucket.com/user/dani_doll98/media/IMG_5474_zpsnwubmh9s.jpg.html?sort=3&o=2

Answer 1

The graph represents five cities and the routes between them.

(a)

Inspector must inspect each road on the map, From Pottsville making only one trip along each road and then fly out of Tinkertown .

Route map can be decided by the Euler's path theorem.

Euler's path  is a path that doesnot have back to track or pass by the same road twice.

Euler's path theorem:  

Euler's path theorem states that if a graph has exactly two vertices of Odd degree, then it has an Euler path that starts and ends on the Odd degree vertices.

Observe the graph:

Degree of each vertices are and .

From the Eulers theorem, graph has two vertices of odd degree, then it has an Euler's path that starts with Pottsville  and ends with Tinkertown .

Route map can be .