Seven Bridges of Königsberg
Next, Euler observed that (except at the endpoints of the walk), whenever one enters a vertex by a bridge, one leaves the vertex by a bridge. In any walk of the graph, the number of times one enters a vertex equals the number of times one leaves it. If every bridge has been crossed exactly once, it follows that, for each land mass (except for the ones chosen for the start and finish), the number of bridges touching that land mass must be even. This is because if there are n bridges, it is crossed exactly 2n times. However, all four of the land masses in the original problem are touched by an odd number of bridges (one is touched by 5 bridges, and each of the other three are touched by 3). There are at most two masses which can be the endpoints of a walk. So the proposition of a walk crossing each bridge once leads to a contradiction.