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Suppose we are in the house one the left-hand side. Then it should be possible to go via a green connection and the green electricity supply to the house in the middle. It should also be possible to go via a blue connection and the water supply to the house on the right-hand side. And of course one can also go via a red connection and the gas supply from the house on the right side to the house on the left side.
This is sketched in the figure on the right-hand side.
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For clarity, we reshape the figure above a little bit in such a way that the houses and the supplies are ending up in a nice circle. The
connections in the new figure are the same as in the figure above. Now there still has to be layed a green, a red and a blue connection.
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There are only two possibilities to lay down these three additional connections. The first possibility is through the innner of the circle, the other one is through the outside. If the green connection is layed through the inner and the blue through the outer, the situation is as in the figure on the right-hand side. But now it is clear that the red connection never can be layed down without crossing. If it was layed down via the outside of the circle it crosses the blue connection, and if it was layed down via the inner it crosses the green connection.
Hence it is impossible to lay down the connections in such a way that they do not cross each other.
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It is however possible, if you take the risk of stealing by the neighbors. Then you for example lay down the red connection below a house like in the figure.
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