Königsberg had seven bridges connecting two islands and two riverbanks. People asked: can you walk a route that crosses every bridge exactly once? Euler proved you can't.
Königsberg's seven bridges joined two islands and two riverbanks. Can you take a walk that crosses every bridge exactly once?
Count the odd corners
Turn the city into dots (land) and lines (bridges). A walk that uses every line once needs every dot to have an even number of lines — one in, one out, paired up — apart from at most two dots where you're allowed to start and stop. Königsberg had four odd dots, so it failed by two.
Euler invented an entire branch of mathematics — graph theory — just to answer 'can I walk this?' Today that same idea routes delivery trucks, plans snowplough rounds, and checks circuit boards.
Can you trace a square with both diagonals drawn in, without lifting your pen or repeating a line?
Why not
If a path enters and leaves a place an equal number of times, the bridges connecting it must be even (entrance + exit pairs). All four landmasses had odd numbers of bridges. So no — impossible.
This is the founding problem of graph theory. Euler invented a whole branch of maths to answer 'can I walk this?'.