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THE UNREASONABLE UTILITY OF RECREATIONAL MATHEMATICS
by Prof. David Singmaster
For First European Congress of Mathematics, Paris, July, 1992. Amended
on 24 Jan 1993 and 7 Sep 1993.
Les hommes ne sont jamais plus ingnieux que dans l'invention des jeux.
[Men are never more ingenious than in inventing games.]
Leibniz to De Montmort, 29 Jul 1715.
Amusement is one of the fields of applied mathematics.
William F. White; A Scrap-Book of Elementary Mathematics; 1908.
... it is necessary to begin the Instruction of Youth with the
Languages and Mathematicks. These should ... be taught to-gether, the
Languages and Classicks as ... Business and the Mathematicks as ...
Samuel Johnson, first President of Columbia University, in 1731.
My title is a variation on Eugene Wigner's famous essay 'The
unreasonable effectiveness of mathematics in the physical sciences'. Like
Wigner, I originally did not come up with any explanation, but more recently
I have begun to formulate an explanation. But first let me describe the
background and illustrate the situation.
For a decade, I have been working to find sources of classical problems
in recreational mathematics. This has led to an annotated
bibliography/history of the subject, now covering about 392 topics on about
456 pages. (404 topics on 500 pp)
1. THE NATURE OF RECREATIONAL MATHEMATICS
To begin with, it is worth considering what is meant by recreational
mathematics. An obvious definition is that it is mathematics that is fun,
but almost any mathematician will say that he enjoys his work, even if he is
studying eigenvalues of elliptic differential operators, so this definition
would encompass almost all mathematics and hence is too general. There are
two, somewhat overlapping, definitions that cover most of what is meant by
First, recreational mathematics is mathematics that is fun and popular
- that is, the problems should be understandable to the interested layman,
though the solutions may be harder. (However, if the solution is too hard,
this may shift the topic from recreational toward the serious - e.g.
Fermat's Last Theorem, the Four Colour Theorem or the Mandelbrot Set.)
Secondly, recreational mathematics is mathematics that is fun and used
as either as a diversion from serious mathematics or as a way of making
serious mathematics understandable or palatable. These are the pedagogic
uses of recreational mathematics. They are already present in the oldest
known mathematics and continue to the present day.
Mathematical recreations are as old as mathematics itself. The
earliest piece of Egyptian mathematics, the Rhind Papyrus of c-1800, has a
problem (No. 79 - OHPs) where there are 7 houses, each house has 7 cats,
each cat ate 7 mice, each mouse would have eaten 7 ears of spelt and each
ear of spelt would produce 7 hekat of spelt. Then 7 + 49 + 343 + 2401 +
16807 is computed. A similar problem of adding powers of 7 occurs in
Fibonacci (1202), in a few later medieval texts and in the children's riddle
rhyme "As I was going to St. Ives". Despite the gaps in the history it is
tempting to believe that "St. Ives" is a descendent from the ancient
Egyptians. Though there is some question as to whether this problem is
really a fanciful exercise in summing a geometric progression, it has no
connection with other problems in the papyrus and seems to be inserted as a
diversion or recreation.
The earliest mathematical works from Babylonia also date from about
-1800 and they include such problems as the following on AO 8862 (OHP) "I
know the length plus the width of a rectangle is 27, while the area plus the
difference of the length and the width is 183. Find the length and width."
By no stretch of the imagination can this be considered a practical problem
- rather it is a way of presenting two equations in two unknowns which
should make the problem more interesting for the student.
These two aspects of recreational mathematics - the popular and the
pedagogic - overlap considerably and there is no clear boundary between them
and "serious" mathematics. In addition, there are two other independent
fields which contain much recreational mathematics: games and mechanical
Games of chance and games of strategy also seem to be about as old as
human civilization. The mathematics of games of chance began in the Middle
Ages and its development by Fermat and Pascal in the 1650s rapidly led to
probability theory and insurance companies based on this theory were founded
in the mid-18C. The mathematics of games of strategy only started about the
beginning of the 20th century, but soon developed into game theory.
Mechanical puzzles range widely in mathematical content. Some only
require a certain amount of dexterity; others require ingenuity and logical
thought; while others require systematic application of mathematical ideas
or patterns, such as Rubik's Cube, the Chinese Rings, the Tower of Hanoi,
The creation of beauty often leads to questions of symmetry and
geometry which are studied for their own sake - e.g. the carved stone balls.
This outlines the conventional scope of recreational mathematics, but
there is some variation due to personal taste.
2. THE UTILITY OF RECREATIONAL MATHEMATICS
How is recreational mathematics useful?
Firstly, recreational problems are often the basis of serious
mathematics. The most obvious fields are probability and graph theory where
popular problems have been a major (or the dominant) stimulus to the
creation and evolution of the subject. Further reflection shows that number
theory, topology, geometry and algebra have been strongly stimulated by
recreational problems. (Though geometry has its origins in practical
surveying, the Greeks treated it as an intellectual game and much of their
work must be considered as recreational in nature, although they viewed it
more seriously as reflecting the nature of the world. From the time of the
Babylonians, algebraists tried to solve cubic equations, though they had no
practical problems which led to cubics.) There are even recreational
aspects of calculus - e.g. the many curves studied since the 16C.
Consequently the study of recreational topics is necessary to understanding
the history of many, perhaps most, topics in mathematics.
Secondly, recreational mathematics has frequently turned up ideas of
genuine but non-obvious utility. I will run through examples of these
Such unusual developments, and the more straightforward developments of
the previous paragraph, demonstrate the historical principle of "The
unreasonable utility of recreational mathematics". This and similar ideas
are the historical and social justification of mathematical research.
Thirdly, recreational mathematics has great pedagogic utility.
A. Recreational mathematics is a treasury of problems which make
mathematics fun. These problems have been tested by generations going back
to about 1800 BC. In medieval arithmetic texts, recreational questions are
interspersed with more straightforward problems to provide breaks in the
hard slog of learning. These problems are often based on reality, though
with enough whimsey so that they have appealed to students and
mathematicians for years. They illustrate the idea that "Mathematics is all
around you - you only have to look for it."
B. "A good problem is worth a thousand exercises." There is no
greater learning experience than trying to solve a good problem.
Recreational mathematics provides many such problems and almost every
problem can be extended or amended. Hence recreational mathematics is also
a treasury of problems for student investigations.
C. Because of its long history, recreational mathematics is an ideal
vehicle for communicating historical and multicultural aspects of
Fourthly, recreational mathematics is very useful to the historian of
mathematics. Recreational problems often are of great age and usually can
be clearly recognised, they serve as useful historical markers, tracing the
development and transmission of mathematics (and culture in general) in
place and time. The Chinese Remainder Theorem, Magic Squares, the Cistern
Problem and the Hundred Fowls Problem are excellent examples of this
process. (The original Hundred Fowls problem, from 5th century China, has a
man buying 100 fowls for 100 cash, roosters cost 5, hens 3 and chicks are 3
for a cash - how many of each did he buy?) The number of topics which have
their origins in China or India is surprising and emphasises our increasing
realisation that modern algebra and arithmetic derive more from Babylonia,
China, India and the Arabs than from Greece.
3. SOME EXAMPLES OF USEFUL RECREATIONAL MATHEMATICS
In this section I will outline a number of examples to show how
recreational mathematics has been useful. (I will stretch recreational a
bit to include some other non-practical topics.)
A. Perhaps the most obvious example is the theory of probability and
statistics which grew from the analysis of gambling bets to the basis of the
insurance industry in the 17th and 18th centuries. Much of combinatorics
likewise has its roots in gambling problems. The theory of Latin squares
began as a recreation but has become an important technique in experimental
B. Greek geometry, though it had some basis in surveying, was largely
an intellectual exercise, pursued for its own sake. The conic sections were
developed with no purpose in mind, but 2000 years later turned out to be
just what Kepler and Newton needed and which now takes men to the moon.
The regular, quasi-regular and Archimedean polyhedra were developed
long before they became the basis of molecular structures. Indeed, the
regular solids are now known to be prehsitoric. Very recently, chemists
have become excited about 'Bucky Balls', carbon structures in various
polyhedral shapes, of which the archetype is the truncated icosahedron, with
60 carbon atoms at the vertices. Such molecules apparently are the basis
for the formation of soot particles in the air. The idea of making such
molecules apparently originated with David Jones, the scientific humorist
who writes as 'Daedalus', in one of his humour columns. Somewhat further in
the past, I recall that chemists produced cubane and dodecane - hydrocarbons
in the shape of a cube and a dodecahedron.
C. Non-Euclidean geometry was developed long before Einstein
considered it as a possible geometry for space.
D. The problem of the Seven Bridges of Knigsberg (OHP), mazes,
knight's tours, circuits on the dodecahedron (Icosian Game) (OHP from 2nd
lecture) were major sources of graph theory and are the basis of major
fields of optimization, leading on to one of the major unsolved problems of
the century: NP = P?? The routes of postmen, streetsweepers and snowplows,
as well as salesmen are worked out by these methods. Further, Hamilton's
thoughts on the Icosian Game led him to the first presentation of a group by
generators and relations. (OHP)
E. Number theory is another of the fields where recreations have been
a major source of problems and these problems have been a major source for
modern algebra. Fermat's Last Theorem lead to Kummer's invention of ideals
and most of algebraic number theory. There was a famous application of
primitive roots to the splicing of telephone cables. Primality and
factorization were traditionally innocuous recreational pastimes, but since
1978 when Rivest, Shamir and Adleman introduced their method of public-key
cryptography, my friends in this field get rung up by reporters wanting to
know if the national security is threatened. The factorization of a big
number or the determination of the next Mersenne prime are generally front
page news now.
F. A major impetus for algebra has been the solving of equations.
The Babylonians already gave quadratic problems where the area of a
rectangle was added to the difference between the length and the width.
This clearly had no practical significance. Similar impractical problems
led to cubic equations and the eventual solution of the cubic. Negative
solutions first become common in medieval puzzle problems about men buying a
horse or finding a purse.
Galois fields and even polynomials over them are now standard tools for
G. Even in analysis, the study of curves (e.g. the cycloid) had some
H. Topology has much of its origins in recreational aspects of curves
and surfaces. Knots, another field once generally considered of no possible
use, are now of great interest to molecular biologists who have discovered
that DNA molecules form into closed chains which may be knotted, or not
The Mbius strip arose about 1858 in work by both Mbius and Listing,
Listing being apparently a bit earlier, though a five twist strip may occur
in Roman mosaics. (OHPs) By 1890, it was already being used as a magic
trick - magic being another application of mathematics - indeed some people
view all mathematics as magic! More recently, such strips have served as
the basis of works by M. C. Escher - art being yet another application of
mathematics. The Mbius strip has also been patented several times! - e.g.
as a single-sided conveyor belt which has double the wearing surface.
(OHPs) None of the patents that I have seen make any reference to any
previous occurrence of the concept. Gardner says it has also been patented
as a non-inductive resistor. Those with dot matrix printers, etc., may (or
may not) know that printer ribbons commonly have a twist so they are Mbius
strips in order to allow the printer to use both edges. I first discovered
this when I found one of our technicians trying to put such a ribbon back
into its cartridge - he had done it several times and it kept coming out
twisted which he thought was his mistake!
I. In combinatorics, the pattern of the Chinese Rings puzzle is the
binary coding known as the Gray Code, patented as an error-minimising code
by Frank Gray of Bell Labs in 1953 and already used in the same way by
Baudot in the 1870s.
I would like to present another binary coding which Baudot utilized.
Chain codes = memory wheels.
THE PENROSE PIECES
Penrose's Pieces have led to the discovery of a new kind of solids -
I will only sketch the ideas here, with some references.
South Bank Polytechnic's coat of arms included 'the net of half a
dodecahedron', i.e. a pentagon surrounded by five other pentagons. (OHP)
One of the basic results of crystallography is that no crystal structure can
have five-fold symmetry. In 1973, I wrote to Roger Penrose on a Polytechnic
letterhead which shows the half dodecahedron. Penrose had long been
interested in tiling the plane with pieces that could not tile the plane
periodically and the letterhead inspired him to try to fill the plane with
pentagons and other related shapes. He soon found such a tiling with six
kinds of shape (OHP) and then managed to reduce it to two shapes which could
tile the plane in uncountably many ways, but in no periodic way. (OHP)
Some of the tilings have a five-fold centre of symmetry, and all have a sort
of generalised five-fold symmetry. They are now called 'quasicrystals'.
These tilings fascinated both geometers and crystallographers and were
extensively studied from the mid-1970s. Penrose's 'kites and darts' shapes
were simplified further to 'fat and thin rhombuses' (OHP) and extended to
three dimensions where they are related to the rhombic triacontahedron
(OHPs). Though the tilings are not periodic, they have quasi-axes and
quasi-planes, which can cause diffraction. (OHPs) Using these,
crystallographers determined the diffraction pattern which a hypothetical
quasicrystal would produce - it has a ten-fold centre of symmetry. In 1984,
such diffraction patterns were discovered by Shechtman in a sample of
rapidly cooled alloy now known as Shechtmanite and some 20 substances are
now known to have quasi-crystalline forms. Indeed, examples were found
about 30 years earlier but the diffraction patterns were discarded as being
erroneous! It is not yet known whether such materials will be useful but
they may be harder or stronger than other forms of the alloys and hence may
find use on aeroplanes, rockets, etc. So a mathematical flight of fancy has
led to the discovery of a new kind of matter on which we may be flying in
[See Scientific American for January 1979 and August 1986 for
expositions of this topic.]
If there is time, I will cover the following as a further utility.
An additional utility of recreational mathematics is that it provides
us a way to communicate mathematical ideas to the public at large.
Mathematicians tend to underestimate the public interest in mathematics.
[Lee Dembart of the Los Angeles Times wrote that when he told people he was
going to a conference on recreational mathematics, they replied that it was
a contradiction in terms! And we all know the social situation when you
confess that you are a mathematician and the response is "Oh. I was never
any good at maths."] Yet somewhere approaching 200 million Rubik Cubes were
sold in three years! Indeed there have been more Rubik Cubes sold in
Hungary than there are people. The best known example of a best-selling
game is Monopoly which has taken 50 years to sell about 90 million examples.
Another measure of the popularity of recreational mathematics is the
number of books that appear in the field each year - perhaps 50 in English
alone. The long term best-seller in English must be Ball's Mathematical
Recreations and Essays now in its 101st year and its 13th edition. It has
rarely been out of print in that time. And there are many older books, such
as Bachet's book of 1612 which had three editions in the late 19C, the last
of which has been reprinted several times in this century.
Many newspapers and professional magazines run regular mathematical
puzzles, though this was more common in the past. Henry Dudeney published
weekly columns for about 15 years and then monthly columns for about 20
years. Martin Gardner's columns were a major factor in the popularity of
Scientific American and probably inspired more students to study mathematics
than any other influence. I have heard that circulation dropped
significantly when he retired. Other major names in the field are the
following. In English: Lewis Carroll, Sam Loyd, Professor Hoffmann, Hubert
Phillips, Tom O'Beirne, Douglas Barnard. In German: Wilhelm Ahrens, Hermann
Schubert, Walther Lietzmann. In French: douard Lucas, Pierre Berloquin.
[I am now trying to carry on this tradition by contributing to the Daily
Telegraph and the new magazine Focus.]
There really is considerable interest in mathematics out there and if
we enjoy our subject, it should be our duty and our pleasure to try to
encourage and feed this interest. Indeed, it may be necessary for our self-
WHY IS RECREATIONAL MATHEMATICS SO USEFUL?
As I said earlier, I have only a tentative answer to this, but it also
partly answers Wigner's question. Mathematics has been described as a
search for pattern - and that certainly describes much of what we do and
also much of what most scientists do. But how do we find patterns? The
real world is messy and patterns are difficult to see. As we begin to see a
pattern, we tend to remove all the inessential details and get to an ideal
or model situation. These models may be so removed from reality that they
become fanciful or even recreational. E.g. physicists deal with
frictionless perfectly elastic particles, weightless strings, ideal gases,
etc. Then such models get modified and adapted into a large variety of
models. Now one of the ways in which a science progresses is by seeing
analogies between reality and simpler situations. E.g. the idea of the
circulation of the blood could not be developed until the idea of a pump was
known and somewhat understood. The behaviour of a real system cannot be
developed until one can see simpler models within it. But what are these
simpler models? They are generally among the large variety of models which
have been created in the past, often recreational or fanciful. Perhaps the
clearest example is graph theory, where Euler made a simple model of the
reality that he was studying, then later workers found that model useful in
other situations. Thus recreational mathematics helps as a major source of
mathematical models, which are the raw material for mathematical research
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