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NSA's 2018 Puzzle Periodical

Try your hand at this month's problem written by a member of our expert workforce.

August 2018 Puzzle Periodical - Crop Calculations

Groups of Ring Fields

Creator Challenge Difficulty Rating: Medium

Puzzle created by: Dr. James M., Operations Researcher

Problem

A farmer, known for his unusual aesthetics, planted five circular fields. Four of the circles have a one furlong radius and were arranged so their centers form a square and they are each tangent to two of the other three circles. The fifth field was planted in the area enclosed by the four circles such that it is tangent to all of them.

What is the total area of the unplanted land enclosed by the five circles?

In this image, the fields are shown in green. We are interested in measuring the red regions.

Diagram of five circular fields. Four of the circles have a one furlong radius and are arranged so their centers form a square and they are each tangent to two of the other three circles. The fifth field/circle is in the area enclosed by the four circles such that it is tangent to all of them.

Click to see the answer!

Solution

Did you calculate the amount of unplanted land enclosed by the five circles?

Our first step is to find the area bounded by the four larger circles. We can then subtract the area of the smaller circle to find the area we are interested in. Let variable r be the radius of the large circles. We know that variable r is equal to one furlong.

The square formed from the centers of the four larger circles has a side length of 2r. Thus it has an area of 4r squared.

Diagram of circular fields with dots marking the center radius of 4 large circles. Lines connect each dot to form a shaded square that covers the smaller 5th circle in the center and the region around it.

If we exclude the region between the four larger circles, we see that we are left with four sectors, each containing a fourth of the area of its respective circle. The area of these regions is then 4 (pie r squared)/4=pie r squared.

Diagram of circular fields with the square area outlined from previous diagram. The shaded region in this diagram only includes the area inside the square but it excludes the small 5th circle and the region surrounding the 5th circle that falls outside of the bodies of the 4 large circles.

Altogether then, the area of the region between the four larger circles is 4r squared - pie r squared = (4 - pie) r squared.

Now we need to find the area of the smaller circle. The centers of any three of the larger circles form a 45-45-90 triangle whose hypotenuse is the diagonal of the square measured earlier. Its length is thus square root of 2 times the side length of the triangle, which we know from before is 2r. Thus the length of the diagonal is 2r square root of 2. If we subtract the lengths of the two radii from this, we get the length of the diameter of the smaller circle. This length is thus 2r square root of 2 - 2r. To get the radius of the smaller circle, we divide this by two, which gives us r(square root of 2 - 1). The area of the smaller circle is then pie[r( square root of 2 - 1)]squared = (3 - 2 square root of 2) pie r squared.

The diagram of 5 circles is shown without any region shaded.  The square has been reduced to a right triangle, but the original 4 points marking the center radius of each large circle are still visible.

We then get the area of the interior region by subtracting these two values, which is (4 - pie) r squared - (3 - 2 square root of 2) pie r squared. Plugging in the value of r=1 furlong gives us the final answer of (4 - pie) - (3 - 2 square root of 2) pie square furlongs, which is about .32 square furlongs.