In 1736 Euler asked whether a stroll could cross each of Königsberg's seven bridges exactly once. He answered it without walking a step — by counting.
Give each district a degree: the number of bridges touching it. Every time a walk enters a district it must leave again, using bridges in pairs — so a district you pass through needs an even degree.
Only the start and the end are allowed to be odd. So a full tour exists exactly when the town has 0 or 2 odd districts (and every bridge sits in one connected piece).
Königsberg has four odd districts — impossible. Pick up the chisel and add one bridge between two odd districts: they turn even, two odd remain, and the tour appears.
The warden's lantern guttered out. The lesson continues without it — everything you have already done is safe.