# Enigmatic Code

Programming Enigma Puzzles

## Puzzle #43: Dividing Grandma’s field

From New Scientist #3266, 25th January 2020 [link]

Two brothers have inherited a plot of land from their grandmother. The map shows that the land is made up of five identical squares, and the green dots indicate the location of four old oak trees.

There are two stipulations in Grandma’s will:

First, the land must be divided so that the brothers get exactly half of the area each, and;
Second, each brother should have two of the trees on their land.

The brothers would love to divide the land with a single straight fence from one edge to another. Can you find a line for the fence that fulfils everyone’s wishes — and without you needing to do any measurement?

[puzzle#43]

### 3 responses to “Puzzle #43: Dividing Grandma’s field”

1. Jim Randell 29 January 2020 at 10:42 am

Here’s one way I thought of:

Suppose the squares are unit squares.

If we were to choose two points along the bottom and top fences and join them with a straight line, then if the distance from the right-hand edge is a for the point on the bottom fence and b for the top fence, the we carve off an area on the right of: (a + b), so if we choose distances such that (a + b) = 5/2 we have successfully divided the field in to two equally sized pieces.

One way to achieve this would be to choose a = 3/4 and b = 7/4.

This line will pass through the exact centre of the upper 3×1 strip (dividing the upper strip exactly in half) and also the exact centre of the lower 2×1 strip (dividing the lower strip exactly in half), and will place two trees in both divisions.

Using some string the centres of the two strips can be determined, and posts placed in these positions. The line of these posts can then be extended to reach the boundary of the field.

• Jim Randell 5 February 2020 at 10:45 am

The official solution gives the same line, and says that the positions of the red crosses “can be found by folding or drawing diagonals so no measurement is required”, although I’m not sure how one can fold a field.

2. Bob Hughes 20 February 2020 at 5:28 am

Thanks Jim, I missed the New Scientist issue with the answer.

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