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.
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.
But now consider Problem 2, with the arrangement of appliances where the positions of ( b ) and ( c ) have been switched:
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.
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?
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:
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.
This is a very old problem. It was used as an example by Ian Stewart in his Foreword to George Pólya’s famous book:
Pólya, George (1945): How to Solve It. Princeton University Press. Penguin Books edition (1990) with Foreword by Ian Stewart.
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