Connecting Three Houses With Gas, Electricity and Water Without Crossings
Suppose we need to connect three houses 1, 2 and to electicity E, gas G and water E, with the pipes/cables not allowed to cross.
Connect house 1 to electricity, gas and water.
Now connect house 2. Any pipe/cable from 1 to any of E, G, W cannot cross a pipe already drawn. Any routes from 2 to E, G and W must divide the plane into two parts A and B below.
The third house must lie in region A, B or C. If it lies in A there is no route to G or W. If it lies in B there is no route to E or G and if it lies in C there is no route to E.
This means that all three houses cannot be connect to gas, water and electricity without at least one crossing.