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Walls & Lines is a popular classic puzzle that involves a large square divided into five "rooms". The object of the puzzle is to cross each "wall" of the diagram with a continuous line only once.
As with the Seven Bridges of Königsberg, the puzzle may be represented in graphical form with each room corresponding to a vertex (including the outside area as a room) and two vertices joined by an edge if the rooms have a common wall. The resulting multigraph does not contain an Eulerian circuit, which means that this puzzle cannot be solved. Solutions missing one wall, however, are possible (see image).
By changing the rules, a related puzzle could be solved. For instance, by permitting passage through more than one wall at a time (that is, through a corner of a room), or by solving the puzzle on a torus (doughnut) instead of a flat plane.
(note the uncrossed wall – marked with circle)
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