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Sunday Times Teaser – Field for Thought

by Peter Good

Published Sunday August 16 2020 (link)

Farmer Giles had a rectangular field bordered by four fences that was 55 hectares in size. He divided the field into three by planting two hedges, from the mid-point of two fences to two corners of the field. He then planted two more hedges, from the mid-point of two fences to two corners of the field. All four hedges were straight, each starting at a different fence and finishing at a different corner.

What was the area of the largest field when the four hedges had been planted?

2 Comments Leave one →
  1. Brian Gladman permalink

    The layout used is shown above.  Since the answer doesn’t depend on shape of the rectangular field, we can derive the answer by considering it to be a square.  All triangles are similar with non-hypotenuse side ratios of 2:1. Using BC as a unit of length, triangles ABC and ADE are similar with ADE being double the size of ABC. So the side CE of the central square field is 2 and the whole field has a side length AD of \(2\sqrt{5}\). Hence AD/BC = \(1/\sqrt{5}\), which means that the area ratio of the inner and outer squares is \(1/5\) making the central field 11 hectares. The red figures show the relative areas of all the sub-fields.

  2. Tony Smith permalink

    From the diagram: 55×4/20

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