Mrs. Miniver's problem

Mrs. Miniver's problem is a geometry problem about circles. Given a circle A, find a circle B such that the area of the lens formed by intersecting their two interiors is equal to the area of the symmetric difference of A and B (the sum of the area of A  B and the area of B  A).


The problem derives from "A Country House Visit", one of Jan Struther's newspaper articles featuring her character Mrs. Miniver. According to the story:

She saw every relationship as a pair of intersecting circles. It would seem at first glance that the more they overlapped the better the relationship; but this is not so. Beyond a certain point the law of diminishing returns sets in, and there are not enough private resources left on either side to enrich the life that is shared. Probably perfection is reached when the area of the two outer crescents, added together, is exactly equal to that of the leaf-shaped piece in the middle. On paper there must be some neat mathematical formula for arriving at this; in life, none.

Alan Wachtel writes of the problem:

It seems that certain mathematicians took this literary challenge literally, and Fadiman follows it with an excerpt from "Ingenious Mathematical Problems and Methods," by L. A. Graham, who had evidently posed the problem in a mathematics journal. Graham gives a solution by William W. Johnson of Cleveland for the general case of unequal circles. The analysis isn't difficult, but the resulting transcendental equation is messy and can't be solved exactly. When the circles are of equal size, the equation is much simpler, but it still can be solved only approximately.


In the case of two circles of equal size, the ratio of the distance between their centers and their radius is often quoted as approximately 0.807946. However, that actually describes the case when the three areas each are of equal size. The solution for the problem as stated in the story ("when the area of the two outer crescents, added together, is exactly equal to that of the leaf-shaped piece in the middle") is approximately 0.529864.


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