Sunday Times Teaser 2957 – Beyond the Fields We Know
by Stephen Hogg
Published May 26 2019 (link)
Beyond The Fields We Know
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?
Consider a trapezium with a base side of length \(b\), a parallel upper side of length \(c\) and a vertical height \(h\)
If we remove a rectangle of height \(h\) and width \(c\) from the trapezium, we are left with a triangle of height \(h\) and base \(b-c\). This allows us to calculate the area of the whole trapezium:\[A=\left(\frac{b\,- c}{2}\right)h + c\,h = \left(\frac{b+c}{2}\right)h\]The midpoints of the two diagonals are both at a height \(h/2\) so the signpost is also at this height. It is now easy to see that the areas of the lower and upper fields are:\[A_1=\frac{b h}{4}\;\;\;\;A_3 = \frac{c h }{4}\] With the left and right field areas as \(A_2\) and \(A_4\), we hence know that:\[A=2(A_1+A_3)=2(A_2+A_4)\] where one of the fields has an area of \(3\) acres and the opposite field has an area of \(3 – x\) acres with \(x < 1\). So we have:\[A = 12 - 2x \;\;\;\text{with} \;\;\;0 < x < 1\]. Since \(A\) is an integer in acres, the total area of the four fields can only be \(11\) acres.