Euler showed that a network can be traced without lifting a pen when every junction has an even number of connections.

The longer version

Count the connections arriving at each junction. Any path that passes straight through a junction uses two of them, entering and leaving, so only a start or finish may have an unpaired connection. That means a route covering every link exactly once can exist only when there are zero or two junctions with an odd count.

The classic puzzle involved seven bridges across a river in a Prussian city, where all four land masses carried an odd number of links, so no such walk was possible. Analysing it in the eighteenth century is usually taken as the beginning of graph theory. With two odd junctions the walk must begin at one and end at the other.

Where this entry sits

This section collects odd corners of mathematics, from stubborn prime numbers to counting systems older than modern schools. Expect famous constants, probability puzzles, shapes that break everyday intuition, and numbers far too large to picture. The full list sits on the Numbers and Math page, where every entry is listed in order.