## Teaser 2957: Beyond the fields we know

**From The Sunday Times, 26th May 2019** [link]

A field named “Dunsany levels” has four unequal straight sides, two of which are parallel. Anne – with her dog, Newton – walked from one corner of the field straight towards her opposite corner. Leon did the same from an adjacent corner along his diagonal. Yards apart, they each rested, halfway along their paths, where Leon, Anne and a signpost in the field were perfectly aligned. Straight fences from each corner converged at the signpost, making four unequal-area enclosures.

Newton made a beeline for the signpost, on which the whole-number area of the field, in acres, was scratched out. Clockwise, the enclosures were named: “Plunkett’s bawn”, “Three-acre meadow”, “Drax sward” and “Elfland lea”. Anne knew that “Three-acre meadow” was literally true and that “Elfland lea” was smaller by less than an acre.

What was the area of “Dunsany levels” in acres?

[teaser2957]

## Jim Randell 11:03 pm

on24 May 2019 Permalink |The field is a

trapezium(trapezoidin US terminology).By drawing a diagram of the layout we determine that there is only a limited range of possible values for the area of Elfland Lea, and only one of which gives a whole number for the total area of the field.

Solution:The total area of the field is 11 acres.Here is my analysis:

We can draw the trapezium, ABCD, with the parallel sides horizontal, and we get a diagram like this:

I assumed that “halfway” meant exactly at the midpoint.

The midpoints of the diagonals AC and BD are exactly half way between the parallel lines AB and CD, so the signpost lies on the line XY (and obviously within the field).

If we suppose the trapezium has a height of

2h, then the line XY is a distancehfrom both AB and CD.Now if we suppose the signpost S lies somewhere on this line:

If the length of AB is

qand the length of CD isp, and the trapezium has a height of2h, then the area of the trapezium is:And the area of the triangles ABS and CDS are:

(and the exact position of S does not matter, as long as it is on line XY).

So it follows that the combined area of the triangles ADS and BCS is:

(the same as the combined area of triangles ABS and CDS).

And one of these pairs consists of “Three-acre meadow” and “Elfland lea”. So one enclosure is exactly 3 acres, and the other is somewhere between 2 and 3 acres (excluding the endpoints), so the combined area of these two enclosures is between 5 and 6 acres (excluding the endpoints).

The total area of the field is twice this value, so it is an integer between 10 and 12 (excluding the endpoints). The answer follows directly, without the need to write a program.

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