#2             67. A SQUARE DEAL               

The four earliest settlers in the Hardoveering Plain built their houses one mile apart at the vertices of a square. As the years rolled by, and the telephone became a reality, the Avgerinos homestead (A) was linked to the county network by a line from the North-East. It was then decided to connect the remaining three homes. Tenders had been called and three replies received:

      Cowley Cables :     $3000
      Lindus Linesmen : $2828
      Whibley Wiring :    $2732

The town clerk frowned as he sketched on his paper. “We know the cost of telephone cable is fixed at $1000 per mile, so it is easy to reconstruct the plans used by Cowley and Lindus. But I don’t see how Whibley can possibly do the job for that price.”

Do you?
Hints and strategies

Hint 1

Hint 2

Solution

Extensions
HINT 1

Draw the square with its handle, and experiment with different layouts.
HINT 2

Did you think of trying a layout which doesn’t just connect the houses directly? One for example, which has extra linkage points?
SOLUTION

Whibley found the best solution to this shortest distance problem. In this diagram, the angles at the new points X, Y are all 120°.
EXTENSION

This problem is an example of a shortest distance problem. It also sometimes appears as 'the travelling salesman problem' where the salesman wants to minimize the distance he travels, although here there is usually the constraint that the salesman return to his starting point. The added points X and Y are called Fermat points.

You might like to explore these possibilities further by looking up ‘Fermat point’ and ‘travelling saleman problem’ on the web.