After 2000 years of searching we only know of 51 perfect numbers and they're all even. What else do and don't we know about these rare and precious jewels of maths?
Can you move a knight on a chessboard so that it visits every square exactly once? Euler was one of the first to analyse this problem systematically, but some questions about it are still open today.
Can you find a path through on this city map that crosses every bridge exactly once? Euler's answer to this problem started off the filed of graph theory.