(From New Scientist, issue 3429)

Older age does bring some benefits.  My daughters Kate and Laura have offered to help me by taking on the maintenance of my garden, which is rectangular with a small, rectangular vegetable plot in one corner.  The remainder is lawn.

To make it fair for them, I have agreed that my last job in the garden will be to partition it into two with a straight fence, with each daughter getting the same area.

Kate suggested that we forget about the vegetable plot, and only divide the lawn.  She sketched a line on the diagram that would give them each exactly half the lawn (with no awkward pinch point to get the mower through).  Laura, meanwhile, drew a fence that would divide the lawn and the vegetable patch into halves.  To make their lines, neither daughter needed to measure anything, they just needed a straight edge.

Can you draw the lines on which Kate and Laura propose to build fences?

Solutions

The lawn part only

Any line dividing a rectangle into two equal parts must go through its centre.  If a single line must divide an area composed of more than one rectangle, it must go through the centres of all.

The centre of a rectangle can be obtained by drawing its diagonals.

The lawn can be thought of as composed of two adjacent rectangles.

But there are two ways to compose the lawn from two rectangles:

In the first one, connecting the centres of the composing rectangles gives the dotted yellow line:

However, that leaves an “awkward pinch point” near the vegetable patch to get the mower through for the daughter looking after the bottom part of the lawn.  So we try the other composition.

Using the second composition it works fine:

Both parts

For dividing both parts the difficult trick is to realise that the composition now consists of the entire garden minus the vegetable patch.  I.e. we need to connect the centre of the entire garden with the centre of the vegetable patch:

Why is this right?

The line divides the vegetable patch into two equal parts, and the entire garden also in two equal parts.  The top part of the garden minus the top part of the patch is the top part of the lawn, and that therefore equals the bottom part of the garden minus the bottom part of the patch.

Remaining question is to see if there are pathological dimensions of garden and patch that make the above reasonings wrong.