Mathigon has a neat introduction to graph theory which includes some fun interactive map problems. The Graphs and Networks lesson requires you to try coloring maps with as few colors as possible and to try to solve Euler's famous Bridges of Königsberg challenge.
What makes Graphs and Networks such a great introduction to graph theory is that it includes a number of interactive map based puzzles which you can try to solve. For example you can try to solve Euler's bridge challenge by tracing a route around the city of Königsberg, crossing each of the city's bridges only once.
The Seven Bridges of Königsberg is a notable problem in mathematics. The river Pregel divides Königsberg into four separate parts. These four parts are connected by seven Bridges. The challenge is to walk around the city, visiting all four parts but only crossing each bridge exactly once. Leonhard Euler's resolution of the problem laid the foundations of graph theory.
Another mathematical problem that involves maps is Map Coloring. For example how many colors do you need to use to color in all the states of the USA when adjacent states can't have the same color. Using the interactive map in Graphs and Networks you can attempt to find out how few colors you need by coloring in all the states in the USA. You can also test your math skills by coloring a number of other interactive maps of different countries.
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