Imagine a room – the Oval Office for example – that has three electrical appliances:
• An air-conditioner ( a ) with an American plug socket ( A ),
• A boiler ( b ) with a British plug socket ( B ),
• A coffee-maker ( c ) with a Continental plug socket ( C ).
The problem is to connect each appliance to the correct socket, avoiding any crossings of the connecting wires.
Fig. 1: Positions of appliances and sockets for Problem 1.
Problem 1
In Problem 1, the positions of the appliances are as shown in Fig. 1 above.
For this distribution of the appliances and sockets, the solution is trivial: connect ( a ) directly to ( A ), ( b ) to ( B ) and ( c ) to ( C ) as shown in Fig. 2.
Fig. 2: Solution to Problem 1
Problem 2
But now consider Problem 2, with the arrangement of appliances where the positions of ( b ) and ( c ) have been switched:
Fig. 3: Positions of appliances and sockets for Problem 2.
We might proceed as before, connecting ( a ) to ( A ) as in Fig. 4, but obviously this makes it impossible to connect the other two appliances without wires crossing.
Fig. 4: Unsuccessful attempt to solve Problem 2.
Before reading further, see if you can find a method of connecting the appliances in Fig. 3 without any crossing wires. Is it possible or impossible?
Topological Thinking
Let us argue topologically: the two problems differ only in the position of the boiler ( b ) and coffee-maker ( c ). Since there is a solution of Problem 1, there should also be a solution of Problem 2.
Imagine transforming the configuration in Problem 1 to that of Problem 2 by continuously moving the boiler ( b ) to the position of the coffee-maker, and the coffee-maker ( c ) to the position of the boiler. The connecting wires would also be moved continuously, avoiding any crossings. It is obvious that a small movement will not render the problem insoluble. The full distortion can be viewed as a homeomorphism, that continuously distorts the region within the room while leaving the positions of the boundary points (and therefore the sockets) unchanged. We would therefore expect Problem 2 to have a solution.
We can see from the following figure how the connections for the boiler and coffee-maker are distorted:
Fig. 5: Distorted connections when ( b ) and ( c ) are interchanged.
It is immediately obvious from Fig. 5 that the air-conditioner ( a ) can be connected to the American socket ( A ) without any crossing of wires.
Fig. 6: Solution to Problem 2.
Sources:
This is a very old problem. It was used as an example by Ian Stewart in his Foreword to George Pólya’s famous book:
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Pólya, George (1945): How to Solve It. Princeton University Press. Penguin Books edition (1990) with Foreword by Ian Stewart.
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