Another name for a simple closed curve is a Jordan curve. The Jordan curve theorem states...

Another name for a simple closed curve is a Jordan curve. The Jordan curve theorem states that every simple closed curve divides the plane into two regions, an inside region and an outside region, and the curve is the boundary between the regions. In order to pass from one region to the other, it is necessary to cross through the curve. Revisit problem from Section and explain how the Jordan curve theorem can be used to show that the problem has no solution.

Problem

Puzzler Henry Dudeney posed a famous problem, presenting the following situation. Suppose that houses are located at points A, B, and C. We want to connect each house to water, electricity, and gas located at points W, G, and E without any of the pipes or wires crossing each other, (NOTE: This problem is related to a branch of mathematics called graph theory.)

a. Try making all of the connections. Can it be done? If so, how?

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b. Suppose that the owner of one of the houses, say B, is willing to let the pipe for one of his neighbors’ connections pass through his house. Then can all of the connections be made? If so, how?