Sixth Book of Mathematical Diversions
Ronald GrahamTen years ago the writer of a mathematics
textbook would have been considered
frivolous by his colleagues if his book in-
cluded puzzles and other entertaining
topics. This is no longer true. Exercises in
the first two volumes of Donald E. Knuth's
monumental work in progress, The Art
of Computer Programming (Reading:
Addison-Wesley, 1968, 1969), are filled
with recreational material. There are even
textbooks in which a recreational emphasis
is primary. A delightful instance is Harold
R. Jacobs's Mathematics: A Human En-
deavor, subtitled A Textbook for Those
Who Think They Don't Like the Subject
(San Francisco: W. H. Freeman and Co.,
1970). Richard Bellman, Kenneth L. Cooke,
and Jo Ann Lockett, authors of Algorithnzs,
Graphs, and Computers (New York: Aca-
demic Press, 1970), write in their preface,
"The principal medium we have chosen to
achieve our goals is the mathematical
puzzle."
The trend is not hard to understand. It
is part of the painfully slow recognition by
educators that students learn best who are
motivated best. Mathematics has never
been a dreary topic, although too often it
has been taught in the dreariest possible
way. There is no better way to relieve the
tedium than by injecting recreational top-
ics into a course, topics strongly tinged
with elements of play, humor, beauty, and
surprise. The greatest mathematicians al-
ways looked upon their subject as a source
of intense intellectual delight and seldom
hesitated to pursue problems of a recre-
ational nature. If you flip the leaves of
W. W. Rouse Ball's classic British work,
Mathematical Recreations and Essays
(first published by Macmillan in 1892 and
soon to be issued in a twelfth revised edi-
tion), you will find the names of celebrated
mathematicians on almost every page.
Euclid himself, among the earliest of
the mathematical giants, wrote an entire
book (unfortunately it did not survive) on
geometrical fallacies. This is a topic cov-
ered in standard works on recreational
mathematics but curiously avoided in most