Teaser 2485: [Circular field]
From The Sunday Times, 9th May 2010 [link] [link]
Jack and Katy have inherited a circular field, with North Gate (N) at the northernmost point and East Gate (E), South Gate (S) and West Gate (W) at the appropriate points.
Jack’s share of the field is one hectare in area. For each point P in his share, [exactly] three of the angles NPE, EPS, SPW and WPN are acute. For each point in Katy’s share, however, fewer than three of those angles are acute.
How far is it between North Gate and East Gate?
This puzzle was originally published with no title.
[teaser2485]
S2T2 
Jim Randell 7:32 am on 22 August 2025 Permalink |
If we draw a circle and consider the angle subtended by the diameter with any point P, then:
So we can draw circles with diameters of NE, ES, SW, WN, and then Jack’s region is any point that is outside exactly 3 of the circles, and Katy’s region is any point that is outside fewer than 3 of the circles.
The situation is this:
Jack’s area (blue) consists of points that are outside exactly 3 of the circles, and Katy’s area (green) consists of points that are outside exactly 2 of the circles.
If we suppose the distance we are interested in is 2d:
Then the area of the semi-circle with diameter NE, radius = d is: (1/2) 𝛑 d^2.
And this area is the same as that of the triangle NOE and two half-petals (= 1 petal):
And Katy’s area (K) consists of 4 petals:
The radius of the entire circular field is: OE = d √2.
And so the total area of the field is then: 2 𝛑 d^2.
And so Jack’s area (J) is:
And this is equal to 1 hectare = 10_000 sq m.
And the distance we are interested in is NE = 2d, hence:
Solution: The distance between the N gate and E gate is 100 m.
And Katy’s area is:
So, only 57% the size of Jack’s area.
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