Three neighbours buy a piece of land that they want to cultivate as a garden. The land has the shape of a square. To avoid that their petunias and pumpkins get in the way of each other, they decide to split the garden into three cells of equal area. To keep things simple, the border between two adjacent cells should be a straight line. Under these constraints, can you help them to divide their garden such that the total length of the fence is minimized? How would you divide the garden for five, six, seven or eight neighbours?

In minimum fencing problems like this there should be a central point where the three lines meet at 120-degree angles. Soap bubbles stretched over a frame in a 3-D simulation of a 2-D problem will seek this 120-degree angle, or rather, set of angles.

So the only determination to be made is where this central meeting point is, and what orientation the three vanes (fences) have.

Posted by Charlie on 2006-12-27 14:50:58