Puzzle #213: Cross sward
From New Scientist #3430, 18th March 2023 [link] [link]
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?
See also: Puzzle #43.
If we place a point in the centre of a rectangular area, then any straight line passing through that point will divide the rectangle in half.
So this gives us a way to find a candidate for Laura’s fence. We place one post in the centre of the vegetable plot, and another in the centre of the entire plot, and then run the fence through both of them.
As the line goes through the centre of the entire plot each daughter gets an area that is half of the entire plot, and as it also goes through the centre of the vegetable plot each daughter gets an area that is half of the vegetable plot. As the area of the lawn is the area of the entire plot minus the area of the vegetable plot, each daughter gets half of the lawn.
For Kate we can divide just the lawn into 2 rectangular plots, and make line joining their centres. One obvious way is to use the small rectangle above the vegetable plot and the remaining large rectangle of lawn, but this will create a narrow “pinch point”. Another way is to extend the small rectangle horizontally to give a strip the full width of the plot, and have a narrower rectangular plot below.
I haven’t seen the official solution. I have to confess this puzzle totally defeated me.
The problem with the above solution is determining the positions of the posts. Practically, if you stand at one corner of the plot and look along to the opposite corner, you could get someone to mark a short line along that line about where the centre is, by calling out instructions. Then repeat that with other pair of corners. Where the 2 lines cross is where the post goes.
That works for Laura’s solution. For Kate’s you would have to mark the points on the edge of the lawn where the edges of the vegetable plot extended meet the edges of the lawn and use those as the ‘corner points’ as above.
@Mike: The puzzle says they sketched the lines on the diagram, so I think that would be fairly easy to do.
Although equipped with some string and pegs I think it would also be relatively straightforward to do on the ground too.
I note the puzzle does say they sketched the lines on the diagram, not on the plot, which is what I was thinking. So I agree that makes the problem fairly simple.
I should have tried to solve the puzzle in NS and not the one in my head.