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?

To at least establish a boundary, I formulated a series of rectangular answers to the problem.

2 regions:

Trivial division into two rectangles of 1/2 x 1

Total border length = 1

3 regions:

Divide the square into rectangles of 1/3 x 1 and 2/3 x 1. Then split the latter into two rectangles of 1/2 x 2/3.

Total border length = 1+2/3 = 1.6667

4 regions:

Trivial division into four rectangles of 1/2 x 1/2.

Total border length = 2

5 regions:

Divide the square into rectangles of 2/5 x 1 and 3/5 x 1. Then split the first large rectangle into two rectangles of 2/5 x 1/2. Similarly split the second large rectangle into three rectangles of 1/3 x 3/5.

Total border length = 2+3/5 = 2.6

6 regions:

Trivial division into four rectangles of 1/3 x 1/2.

Total border length = 3

7 regions:

Divide the square into two rectangles of 2/7 x 1 and one rectangle of 3/7 x 1. Then split the first two rectangles each into two smaller rectangles of 2/7 x 1/2. Similarly split the third large rectangle into three rectangles of 1/3 x 3/7.

Total border length = 3+3/7 = 3.4286

8 regions:

Divide the square into two rectangles of 3/8 x 1 and one rectangle of 1/4 x 1. Then split the first two rectangles each into three smaller rectangles of 3/8 x 1/3. Similarly split the third large rectangle into two rectangles of 1/4 x 1/2.

Total border length = 3+3/4 = 3.75

9 regions:

Trivial division into nine rectangles of 1/3 x 1/3.

Total border length = 4