This document discusses the unreasonable utility of recreational mathematics. It begins by defining recreational mathematics as mathematics that is fun and popular, understood by laypeople, or used for pedagogical purposes. The document then argues that recreational mathematics has been useful in several ways: it has stimulated serious mathematics; provided problems for students; communicated history and culture; and aided historians. Several examples are provided to illustrate how recreational problems led to probability, graph theory, geometry concepts used in science, and more.

1. the serious - e.g. Fermat's Last Theorem, the Four Col-
THE UNREASONABLE UTILITY OF RECREA-
our Theorem or the Mandelbrot Set.)
TIONAL MATHEMATICS
Secondly, recreational mathematics is mathe-
by Prof. David Singmaster
matics that is fun and used as either as a diversion from
serious mathematics or as a way of making serious
For First European Congress of Mathematics, Paris,
mathematics understandable or palatable. These are the
July, 1992. Amended on 24 Jan 1993 and 7 Sep 1993.
pedagogic uses of recreational mathematics. They are
already present in the oldest known mathematics and
Les hommes ne sont jamais plus ing‚nieux que
continue to the present day.
dans l'invention des jeux. [Men are never more ingen-
Mathematical recreations are as old as mathe-
ious than in inventing games.]
matics itself. The earliest piece of Egyptian mathemat-
Leibniz to De Montmort, 29 Jul 1715.
ics, the Rhind Papyrus of c-1800, has a problem (No. 79
- OHPs) where there are 7 houses, each house has 7 cats,
Amusement is one of the fields of applied
each cat ate 7 mice, each mouse would have eaten 7 ears
mathematics.
of spelt and each ear of spelt would produce 7 hekat of
William F. White; A Scrap-Book of Elemen-
spelt. Then 7 + 49 + 343 + 2401 + 16807 is computed.
tary Mathematics; 1908.
A similar problem of adding powers of 7 occurs in Fibo-
nacci (1202), in a few later medieval texts and in the
... it is necessary to begin the Instruction of
children's riddle rhyme quot;As I was going to St. Ivesquot;.
Youth with the Languages and Mathematicks. These
Despite the gaps in the history it is tempting to believe
should ... be taught to-gether, the Languages and
that quot;St. Ivesquot; is a descendent from the ancient Egyp-
Classicks as ... Business and the Mathematicks as ... Di-
tians. Though there is some question as to whether this
version.
Samuel Johnson, first President of Columbia problem is really a fanciful exercise in summing a geo-
University, in 1731. metric progression, it has no connection with other prob-
lems in the papyrus and seems to be inserted as a diver-
My title is a variation on Eugene Wigner's fa- sion or recreation.
The earliest mathematical works from Babylo-
mous essay 'The unreasonable effectiveness of mathe-
nia also date from about -1800 and they include such
matics in the physical sciences'. Like Wigner, I origi-
problems as the following on AO 8862 (OHP) quot;I know
nally did not come up with any explanation, but more
the length plus the width of a rectangle is 27, while the
recently I have begun to formulate an explanation. But
area plus the difference of the length and the width is
first let me describe the background and illustrate the
183. Find the length and width.quot; By no stretch of the
situation.
imagination can this be considered a practical problem -
For a decade, I have been working to find
rather it is a way of presenting two equations in two un-
sources of classical problems in recreational mathemat-
knowns which should make the problem more interest-
ics. This has led to an annotated bibliography/history of
ing for the student.
the subject, now covering about 392 topics on about 456
These two aspects of recreational mathematics
pages. (404 topics on 500 pp)
- the popular and the pedagogic - overlap considerably
and there is no clear boundary between them and quot;seri-
1. THE NATURE OF RECREATIONAL
ousquot; mathematics. In addition, there are two other inde-
MATHEMATICS
pendent fields which contain much recreational mathe-
matics: games and mechanical puzzles.
To begin with, it is worth considering what is
Games of chance and games of strategy also
meant by recreational mathematics. An obvious defini-
seem to be about as old as human civilization. The
tion is that it is mathematics that is fun, but almost any
mathematics of games of chance began in the Middle
mathematician will say that he enjoys his work, even if
Ages and its development by Fermat and Pascal in the
he is studying eigenvalues of elliptic differential opera-
1650s rapidly led to probability theory and insurance
tors, so this definition would encompass almost all
companies based on this theory were founded in the
mathematics and hence is too general. There are two,
mid-18C. The mathematics of games of strategy only
somewhat overlapping, definitions that cover most of
started about the beginning of the 20th century, but soon
what is meant by recreational mathematics.
developed into game theory.
First, recreational mathematics is mathematics
Mechanical puzzles range widely in mathe-
that is fun and popular - that is, the problems should be
matical content. Some only require a certain amount of
understandable to the interested layman, though the solu-
dexterity; others require ingenuity and logical thought;
tions may be harder. (However, if the solution is too
while others require systematic application of mathe-
hard, this may shift the topic from recreational toward

2. B. quot;A good problem is worth a thou-
matical ideas or patterns, such as Rubik's Cube, the Chi-
sand exercises.quot; There is no greater learning experience
nese Rings, the Tower of Hanoi, Rubik's Clock.
than trying to solve a good problem. Recreational
The creation of beauty often leads to questions
mathematics provides many such problems and almost
of symmetry and geometry which are studied for their
every problem can be extended or amended. Hence rec-
own sake - e.g. the carved stone balls.
reational mathematics is also a treasury of problems for
This outlines the conventional scope of recrea-
student investigations.
tional mathematics, but there is some variation due to
C. Because of its long history, recrea-
personal taste.
tional mathematics is an ideal vehicle for communicat-
ing historical and multicultural aspects of mathematics.
2. THE UTILITY OF RECREATIONAL
MATHEMATICS
Fourthly, recreational mathematics is very use-
ful to the historian of mathematics. Recreational prob-
How is recreational mathematics useful?
lems often are of great age and usually can be clearly
recognised, they serve as useful historical markers, trac-
Firstly, recreational problems are often the ba-
ing the development and transmission of mathematics
sis of serious mathematics. The most obvious fields are
(and culture in general) in place and time. The Chinese
probability and graph theory where popular problems
Remainder Theorem, Magic Squares, the Cistern Prob-
have been a major (or the dominant) stimulus to the
lem and the Hundred Fowls Problem are excellent ex-
creation and evolution of the subject. Further reflection
amples of this process. (The original Hundred Fowls
shows that number theory, topology, geometry and alge-
problem, from 5th century China, has a man buying 100
bra have been strongly stimulated by recreational prob-
fowls for 100 cash, roosters cost 5, hens 3 and chicks are
lems. (Though geometry has its origins in practical sur-
3 for a cash - how many of each did he buy?) The num-
veying, the Greeks treated it as an intellectual game and
ber of topics which have their origins in China or India
much of their work must be considered as recreational in
is surprising and emphasises our increasing realisation
nature, although they viewed it more seriously as reflect-
that modern algebra and arithmetic derive more from
ing the nature of the world. From the time of the Baby-
Babylonia, China, India and the Arabs than from Greece.
lonians, algebraists tried to solve cubic equations,
though they had no practical problems which led to
3. SOME EXAMPLES OF USEFUL REC-
cubics.) There are even recreational aspects of calculus -
REATIONAL MATHEMATICS
e.g. the many curves studied since the 16C. Conse-
quently the study of recreational topics is necessary to
In this section I will outline a number of ex-
understanding the history of many, perhaps most, topics
amples to show how recreational mathematics has been
in mathematics.
useful. (I will stretch recreational a bit to include some
Secondly, recreational mathematics has fre-
other non-practical topics.)
quently turned up ideas of genuine but non-obvious util-
A. Perhaps the most obvious example is
ity. I will run through examples of these later.
the theory of probability and statistics which grew from
Such unusual developments, and the more
the analysis of gambling bets to the basis of the
straightforward developments of the previous paragraph,
insurance industry in the 17th and 18th centuries. Much
demonstrate the historical principle of quot;The unreason-
of combinatorics likewise has its roots in gambling prob-
able utility of recreational mathematicsquot;. This and simi-
lems. The theory of Latin squares began as a recreation
lar ideas are the historical and social justification of
but has become an important technique in experimental
mathematical research.
design.
Thirdly, recreational mathematics has great
B. Greek geometry, though it had some
pedagogic utility.
basis in surveying, was largely an intellectual exercise,
A. Recreational mathematics is a treas-
pursued for its own sake. The conic sections were de-
ury of problems which make mathematics fun. These
veloped with no purpose in mind, but 2000 years later
problems have been tested by generations going back to
turned out to be just what Kepler and Newton needed
about 1800 BC. In medieval arithmetic texts, recrea-
and which now takes men to the moon.
tional questions are interspersed with more straightfor-
The regular, quasi-regular and Archimedean
ward problems to provide breaks in the hard slog of
polyhedra were developed long before they became the
learning. These problems are often based on reality,
basis of molecular structures. Indeed, the regular solids
though with enough whimsey so that they have appealed
are now known to be prehsitoric. Very recently, chem-
to students and mathematicians for years. They illustrate
ists have become excited about 'Bucky Balls', carbon
the idea that quot;Mathematics is all around you - you only
structures in various polyhedral shapes, of which the ar-
have to look for it.quot;

3. have discovered that DNA molecules form into closed
chetype is the truncated icosahedron, with 60 carbon at-
chains which may be knotted, or not knotted.
oms at the vertices. Such molecules apparently are the
The M”bius strip arose about 1858 in work by
basis for the formation of soot particles in the air. The
both M”bius and Listing, Listing being apparently a bit
idea of making such molecules apparently originated
earlier, though a five twist strip may occur in Roman
with David Jones, the scientific humorist who writes as
mosaics. (OHPs) By 1890, it was already being used as
'Daedalus', in one of his humour columns. Somewhat
a magic trick - magic being another application of
further in the past, I recall that chemists produced cu-
mathematics - indeed some people view all mathematics
bane and dodecane - hydrocarbons in the shape of a cube
as magic! More recently, such strips have served as the
and a dodecahedron.
basis of works by M. C. Escher - art being yet another
C. Non-Euclidean geometry was devel-
application of mathematics. The M”bius strip has also
oped long before Einstein considered it as a possible ge-
been patented several times! - e.g. as a single-sided
ometry for space.
D. The problem of the Seven Bridges of conveyor belt which has double the wearing surface.
K”nigsberg (OHP), mazes, knight's tours, circuits on the (OHPs) None of the patents that I have seen make any
dodecahedron (Icosian Game) (OHP from 2nd reference to any previous occurrence of the concept.
lecture) were major sources of graph theory and are the Gardner says it has also been patented as a non-inductive
basis of major fields of optimization, leading on to one resistor. Those with dot matrix printers, etc., may (or
of the major unsolved problems of the century: NP = P?? may not) know that printer ribbons commonly have a
The routes of postmen, streetsweepers and snowplows, twist so they are M”bius strips in order to allow the
as well as salesmen are worked out by these methods. printer to use both edges. I first discovered this when I
Further, Hamilton's thoughts on the Icosian Game led found one of our technicians trying to put such a ribbon
him to the first presentation of a group by generators and back into its cartridge - he had done it several times and
relations. (OHP) it kept coming out twisted which he thought was his mis-
E. Number theory is another of the take!
I. In combinatorics, the pattern of the
fields where recreations have been a major source of
Chinese Rings puzzle is the binary coding known as the
problems and these problems have been a major source
Gray Code, patented as an error-minimising code
for modern algebra. Fermat's Last Theorem lead to
by Frank Gray of Bell Labs in 1953 and already used in
Kummer's invention of ideals and most of algebraic
the same way by Baudot in the 1870s.
number theory. There was a famous application of
I would like to present another binary coding
primitive roots to the splicing of telephone cables. Pri-
which Baudot utilized. Chain codes = memory wheels.
mality and factorization were traditionally innocuous
recreational pastimes, but since 1978 when Rivest,
THE PENROSE PIECES
Shamir and Adleman introduced their method of public-
key cryptography, my friends in this field get rung up by
Penrose's Pieces have led to the discovery of a
reporters wanting to know if the national security is
new kind of solids - the 'quasicrystals'.
threatened. The factorization of a big number or the de-
I will only sketch the ideas here, with some
termination of the next Mersenne prime are generally
references.
front page news now.
South Bank Polytechnic's coat of arms in-
F. A major impetus for algebra has
cluded 'the net of half a dodecahedron', i.e. a pentagon
been the solving of equations. The Babylonians already
surrounded by five other pentagons. (OHP) One of the
gave quadratic problems where the area of a rectangle
basic results of crystallography is that no crystal struc-
was added to the difference between the length and the
ture can have five-fold symmetry. In 1973, I wrote to
width. This clearly had no practical significance. Simi-
Roger Penrose on a Polytechnic letterhead which shows
lar impractical problems led to cubic equations and the
the half dodecahedron. Penrose had long been interested
eventual solution of the cubic. Negative solutions first
in tiling the plane with pieces that could not tile the
become common in medieval puzzle problems about
plane periodically and the letterhead inspired him to try
men buying a horse or finding a purse.
Galois fields and even polynomials over them to fill the plane with pentagons and other related shapes.
are now standard tools for cryptographers. He soon found such a tiling with six kinds of shape
G. Even in analysis, the study of curves (OHP) and then managed to reduce it to two shapes
(e.g. the cycloid) had some recreational motivation. which could tile the plane in uncountably many ways,
H. Topology has much of its origins in but in no periodic way. (OHP)
Some of the tilings have a five-fold centre of symmetry,
recreational aspects of curves and surfaces. Knots, an-
and all have a sort of generalised five-fold symmetry.
other field once generally considered of no possible use,
They are now called 'quasicrystals'. These tilings fasci-
are now of great interest to molecular biologists who

4. columns for about 15 years and then monthly columns
nated both geometers and crystallographers and were
for about 20 years. Martin Gardner's columns were a
extensively studied from the mid-1970s. Penrose's 'kites
major factor in the popularity of Scientific American and
and darts' shapes were simplified further to 'fat and thin
probably inspired more students to study mathematics
rhombuses' (OHP) and extended to three dimensions
than any other influence. I have heard that circulation
where they are related to the rhombic triacontahedron
dropped significantly when he retired. Other major
(OHPs). Though the tilings are not periodic, they have
names in the field are the following. In English: Lewis
quasi-axes and quasi-planes, which can cause diffrac-
Carroll, Sam Loyd, Professor Hoffmann, Hubert Phil-
tion. (OHPs) Using these, crystallographers determined
lips, Tom O'Beirne, Douglas Barnard. In German:
the diffraction pattern which a hypothetical quasicrystal
Wilhelm Ahrens, Hermann Schubert, Walther
would produce - it has a ten-fold centre of symmetry. In
1984, such diffraction patterns were discovered by Lietzmann. In French: douard Lucas, Pierre Berloquin.
Shechtman in a sample of rapidly cooled alloy now [I am now trying to carry on this tradition by contribut-
known as Shechtmanite and some 20 substances are now ing to the Daily Telegraph and the new magazine Fo-
known to have quasi-crystalline forms. Indeed, exam- cus.]
ples were found about 30 years earlier but the diffraction There really is considerable interest in mathe-
patterns were discarded as being erroneous! It is not yet matics out there and if we enjoy our subject, it should be
known whether such materials will be useful but they our duty and our pleasure to try to encourage and feed
may be harder or stronger than other forms of the alloys this interest. Indeed, it may be necessary for our self-
and hence may find use on aeroplanes, rockets, etc. So a preservation.
mathematical flight of fancy has led to the discovery of a
new kind of matter on which we may be flying in the WHY IS RECREATIONAL MATHEMAT-
future! ICS SO USEFUL?
[See Scientific American for January 1979 and
August 1986 for expositions of this topic.] As I said earlier, I have only a tentative answer
If there is time, I will cover the following as a to this, but it also partly answers Wigner's question.
further utility. Mathematics has been described as a search for pattern -
An additional utility of recreational mathemat- and that certainly describes much of what we do and also
ics is that it provides us a way to communicate mathe- much of what most scientists do. But how do we find
matical ideas to the public at large. Mathematicians tend patterns? The real world is messy and patterns are diffi-
to underestimate the public interest in mathematics. cult to see. As we begin to see a pattern, we tend to re-
[Lee Dembart of the Los Angeles Times wrote that when move all the inessential details and get to an ideal or
he told people he was going to a conference on recrea- model situation. These models may be so removed from
tional mathematics, they replied that it was a contradic- reality that they become fanciful or even recreational.
tion in terms! And we all know the social situation E.g. physicists deal with frictionless perfectly elastic
when you confess that you are a mathematician and the particles, weightless strings, ideal gases, etc. Then such
response is quot;Oh. I was never any good at maths.quot;] Yet models get modified and adapted into a large variety of
somewhere approaching 200 million Rubik Cubes were models. Now one of the ways in which a science pro-
sold in three years! Indeed there have been more Rubik gresses is by seeing analogies between reality and sim-
Cubes sold in Hungary than there are people. The best pler situations. E.g. the idea of the circulation of the
known example of a best-selling game is Monopoly blood could not be developed until the idea of a pump
which has taken 50 years to sell about 90 million exam- was known and somewhat understood. The behaviour of
ples. a real system cannot be developed until one can see sim-
Another measure of the popularity of recrea- pler models within it. But what are these simpler mod-
tional mathematics is the number of books that appear in els? They are generally among the large variety of mod-
the field each year - perhaps 50 in English alone. The els which have been created in the past, often recrea-
long term best-seller in English must be Ball's Mathe- tional or fanciful. Perhaps the clearest example is graph
matical Recreations and Essays now in its 101st year and theory, where Euler made a simple model of the reality
its 13th edition. It has rarely been out of print in that that he was studying, then later workers found that
time. And there are many older books, such as Bachet's model useful in other situations. Thus recreational
book of 1612 which had three editions in the late 19C, mathematics helps as a major source of mathematical
the last of which has been reprinted several times in this models, which are the raw material for mathematical re-
century. search
Many newspapers and professional magazines
run regular mathematical puzzles, though this was more
common in the past. Henry Dudeney published weekly