A Question of Sides...

Yesterday's thinking about preconceptions of 3D reminded me of something that I was shown as a child, and which I now know to be a classic puzzle. But armed with 3D thinking, then there's a couple of wrinkles that I'd like to add to the canon...

The premise is simple enough. There are three houses, and three utility suppliers: Electricity, Gas and Water. All you have to do is connect the three houses to each of the utilities, with the one constraint being that you cannot have any of the utility supplies crossing over another - some obscure local bye-law or something is usually part of the patter at this point. The diagram that accompanies the problem is straight-forward:

The start position

People often try connecting one utility to all three houses, and then the second utility, and then the third, two of which are normally easy, but the third is blocked by a previous connection. Back-tracking and re-routing only results in the same blocking, and persistence or ingenuity are no help in finding a solution.

Only the water connection to the middle house to go...

Actually, there isn't a solution, and the mathematics behind it is very well understood, and was neatly captured by Euler in a formula that describes the properties of 'planar graphs' and 'non-planar graphs', of which this is an example. It actually boils down to a planar graph being a collection of points and lines connecting them that can be drawn on a surface without any cross-overs, whilst a non-planar graph is one where you can't. The three houses problem is non-planar, and so can't be solved on a flat surface. Of course, now that you know that you can try to solve it on other surfaces, and a doughnut-shape (or, more formally, a torus) is one example of a surface on which you can solve it.

But one surface which most people forget about is the other side of the piece of paper on which this problem is normally drawn. By using the other side of the piece of paper, it is also possible to solve it. However, one of the consequences of technology is that we have new ways of representing things, and if you draw the diagram on a screen, then an interesting thing happens - you can't use the other side, because:

'Unlike paper, screens don't have an 'other' side.'

Which made me sit down and think hard, because I'd never thought about it before. Screens are single sided, rather like a möebius strip.

If you want to know more about the history and the maths behind this puzzle, then there's an excellent web-page here.