Three utilities problem

The classical mathematical puzzle known as the three utilities problem; the three cottages problem or sometimes water, gas and electricity can be stated as follows:

The problem is an abstract mathematical puzzle which imposes constraints that would not exist in a practical engineering situation.

History

Henry Dudeney states that the problem is "as old as the hills...much older than electric lighting, or even gas"; a review of the problem's history is given by Kullman who states that most published references to the problem characterize it as "very ancient".

Solution

The problem itself, as presented at the top of this article, is strictly impossible on a flat two-dimensional plane. However, a less-strict variation is possible, assuming one is allowed to pass through the cottages and / or utilities. With this alteration, it is "no longer impossible to connect the three cottages with the three different utilities without at least one of the connections crossing another".