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?

I would guess five neighbors would have minimal fencing thusly:

_________________________
|                     |                   |
|                     |                   |
|                     |                   |
|                  /     \                |
|               /           \             |
|            /                 \          |
|------<                     >-------|
|            \                 /          |
|               \           /             |
|                  \     /                |
|                     |                   |
|                     |                   |
|____________|___________|

With the inside sqare having side length sqrt(5)/5 units

Posted by Eric on 2006-12-27 18:17:51