The Prussian city of Konigsberg had seven bridges over its river in the 18th century, and people wondered: could you walk a route that crosses every bridge exactly once? When the mathematician Euler answered that question, an entirely new branch of mathematics — graph theory — was born.
Shrink each landmass to a dot and each bridge to a line, and you get the diagram below. Click the dots in order to cross every bridge exactly once — you cannot reuse a bridge. Try it as many times as you like and see whether it can really be done.
Click any dot to start, then click the next dots in order and try to cross all seven bridges.
Pick a shape and try to draw it in one stroke. See for yourself which shapes work and which never do.
The number of lines meeting at a dot is called its degree. Count how many dots have an odd degree (odd vertices) and you can tell in advance whether a one-stroke drawing is possible.
0 odd vertices → you can start anywhere, draw it in one stroke, and end back where you started (an Euler circuit).
2 odd vertices → it works only if you start at one of them and finish at the other (an Euler path).
3 or more odd vertices → a one-stroke drawing is never possible. Konigsberg had four of them, which is exactly why it could not be done!
Whenever your stroke passes through a dot, it uses one line coming in and one going out, always as a pair. So at any dot that is neither the start nor the finish, each visit consumes two lines — which means its total number of lines (its degree) must be even.
Only two kinds of dot can have an odd number of lines — the start (one extra line leaving) and the finish (one extra line arriving). So there can be at most two odd vertices. With three or more, they cannot all be starts or finishes, and the one-stroke drawing becomes impossible.