Mrs. Miniver's problem is a

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

problem about the

area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while ''surface area'' refers to the area of an open su ...

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...

s. It asks how to place two circles

A

and

B

of given radii in such a way that the

lens A lens is a transmissive optical device which focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a compound lens consists of several simple lenses (''elements'' ...

formed by intersecting their two interiors has equal area to the

symmetric difference In mathematics, the symmetric difference of two sets, also known as the disjunctive union, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets \ and \ is \. T ...

A

and

B

(the area contained in one but not both circles). It was named for an analogy between geometry and

social dynamics Social dynamics (or sociodynamics) is the study of the behavior of groups that results from the interactions of individual group members as well to the study of the relationship between individual interactions and group level behaviors. Overv ...

enunciated by fictional character

Mrs. Miniver ''Mrs. Miniver'' is a 1942 American romantic war drama film directed by William Wyler, and starring Greer Garson and Walter Pidgeon. Inspired by the 1940 novel ''Mrs. Miniver'' by Jan Struther, it shows how the life of an unassuming British ...

, who "saw every relationship as a pair of intersecting circles". Its solution involves a

transcendental equation In applied mathematics, a transcendental equation is an equation over the real (or complex) numbers that is not algebraic, that is, if at least one of its sides describes a transcendental function. Examples include: :\begin x &= e^ \\ x & ...

Origin

The problem derives from "A Country House Visit", one of

Jan Struther Jan, JaN or JAN may refer to: Acronyms * Jackson, Mississippi (Amtrak station), US, Amtrak station code JAN * Jackson-Evers International Airport, Mississippi, US, IATA code * Jabhat al-Nusra (JaN), a Syrian militant group * Japanese Article Numb ...

's newspaper articles appearing in the ''

Times of London ''The Times'' is a British daily national newspaper based in London. It began in 1785 under the title ''The Daily Universal Register'', adopting its current name on 1 January 1788. ''The Times'' and its sister paper ''The Sunday Times'' (fou ...

'' between 1937 and 1939 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.

Louis A. Graham and

Clifton Fadiman Clifton Paul "Kip" Fadiman (May 15, 1904 – June 20, 1999) was an American intellectual, author, editor, radio and television personality. He began his work with the radio, and switched to television later in his career. Background Born in Bro ...

formalized the mathematics of the problem and popularized it among

recreational mathematicians Recreation is an activity of leisure, leisure being discretionary time. The "need to do something for recreation" is an essential element of human biology and psychology. Recreational activities are often done for enjoyment, amusement, or pleasu ...

Solution

The problem can be solved by cutting the lune along the

line segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...

between the two crossing points of the circles, into two

circular segment In geometry, a circular segment (symbol: ), also known as a disk segment, is a region of a disk which is "cut off" from the rest of the disk by a secant or a chord. More formally, a circular segment is a region of two-dimensional space that is ...

s, and using the formula for the area of a circular segment to relate the distance between the crossing points to the total area that the problem requires the lune to have. This gives a

for the distance between crossing points but it can be solved numerically. There are two boundary conditions whose distances between centers can be readily solved: the farthest apart the centers can be is when the circles have equal radii, and the closest they can be is when one circle is contained completely within the other, which happens when the ratio between radii is

\sqrt2

. If the ratio of radii falls beyond these limiting cases, the circles cannot satisfy the problem's area constraint. In the case of two circles of equal size, these equations can be simplified somewhat. The

rhombus In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. Th ...

formed by the two circle centers and the two crossing points, with side lengths equal to the radius, has an angle

\theta\approx 2.605

radians at the circle centers, found by solving the equation

\theta-\sin\theta=\frac,

from which it follows that the ratio of the distance between their centers to their radius is

2\cos\tfrac\theta2\approx0.529864

References

{{reflist, refs= {{citation , last1 = Alsina , first1 = Claudi , last2 = Nelsen , first2 = Roger B. , contribution = 11.8: Mrs. Miniver's problem , isbn = 978-1-4704-5616-0 , pages = 141–142 , publisher = American Mathematical Society , series = Dolciani Mathematical Expositions , title = Icons of Mathematics: An Exploration of Twenty Key Images , title-link = Icons of Mathematics , volume = 56 , year = 2011 {{citation , last = Graham , first = Louis A. , contribution = 3: Mrs. Miniver's problem , contribution-url = https://books.google.com/books?id=xdpQAQAAQBAJ&pg=PA64 , isbn = 978-0-486-28293-0 , pages = 64–66 , publisher = Dover Publications , series = Dover Books on Mathematics , title = Ingenious Mathematical Problems and Methods , year = 1959 {{citation , last = Fadiman , first = Clifton , author-link = Clifton Fadiman , contribution = The Miniver problem , pages = 298–300 , publisher =

Simon & Schuster Simon & Schuster () is an American publishing company and a subsidiary of Paramount Global. It was founded in New York City on January 2, 1924 by Richard L. Simon and M. Lincoln Schuster. As of 2016, Simon & Schuster was the third largest pub ...

, title = The Mathematical Magpie , year = 1962 {{citation , last = Struther , first = Jan , author-link = Jan Struther , access-date = 2022-03-10 , contribution = A Country House Visit , contribution-url = https://digital.library.upenn.edu/women/struther/miniver/miniver.html#12 , publisher = University of Pennsylvania , title = Mrs. Miniver. Originally published as part of a series of columns in ''The Times'', 1937, and in the book ''Mrs. Miniver'', Chatto and Windus, London, 1939. Circles Area Mathematical problems

Origin

Solution

See also

References