A Mathematician's Holiday - Problem 7: Fix the Hotel Rooms.

Requirements:

Pen and paper is all that is required. We also use blow up rubber ring to demonstrate the solution.

Description:
Having got to our hotel rooms Dan, Will and I found that our water, electricity and gas supplies had all been damaged in the fire. Being the helpful souls that we are can we connect each of the utility supplies to each room without crossing the utilities?

Figure 1. Problem set up.

This is a classic mathematical problem known as "The Three Utilities Problem". Critically, it is known to be unsolvable on the flat sheet of the paper. However, the hotel we would be staying in would be three-dimensional and, thus, we are asking the audience to extend their ideas beyond the flat surface.

Note that we really do not go into why the solution is impossible on the flat paper. The reason is because the proof of impossibility depends on some deep ideas from topology. However, the interested reader can find the proof here.

Extension
If we add in another utility, can we still solve the problem on a torus, or doughnut?