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 other words, during any walk in the graph, the number of
times one enters a non-terminal vertex equals the number of times one
leaves it. Now, if every bridge has been traversed 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 (half of them, in the particular traversal, will be traversed
"toward" the landmass; the other half, "away" from it). 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). Since, at most, two land masses can serve as
the endpoints of a putative walk, the proposition of a walk traversing
each bridge once leads to a contradiction.
http://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg#Solutions
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http://www.youtube.com/watch?v=JX3VmDgiFnY
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http://en.wikipedia.org/wiki/File:Tictactoe-X.svg