214 8. NUMBERS AND NUMBER THEORY IN MODERN MATHEMATICS
Bhaskara recognized that there was one more possibility, six short syllables, but
did not mention the possibility of no short syllables. The convention that an empty
product equals 1 was not part of his experience.
Magic squares and Latin squares. In 1274 (see Li and Du, 1987, p. 166) Yang Hui
wrote Xugu Zhaiqi Suanfa (Continuation of Ancient Mathematical Methods for
Elucidating the Strange [Properties of Numbers]), in which he listed magic squares
of order up to 10. According to Biggs (1979, p. 121), there is evidence that the topic
of magic squares had reached a high degree of development in China before this time
and that Yang Hui was merely listing ancient results that were not a topic of current
research, since he seems to have no concept of a general rule for constructing magic
squares. Magic-square-type figures were a source of fascination in Korea, and they
also seem to have spread west from China. Whether from China or as an indigenous
product, magic squares of order up to 6 appear in the Muslim world around the
year 990 (Biggs, 1979, p. 119) and squares of order up to 9 are mentioned. Rules
for constructing such squares were given by the Muslim scholar al-Buni (d. 1225).
From the Muslim world, they entered Europe around the year 1315 in the works of
the Greek scholar named Manuel Moschopoulos, who was claimed as a student by
Maximus Planudes, who was mentioned in Chapter 2. They exerted a fascination
on European scholars also, and the artist Albrecht Diirer incorporated a 4 ÷ 4 magic
square in his famous engraving Melencolia, with the year of its composition (1514)
in the bottom row. The most fascinating thing about them is their sheer number.
Difficult as they are to construct, there are nevertheless 880 distinct 4x4 magic
squares.
Magic squares occur profusely throughout Indian, Chinese, and Japanese math-
ematics, alongside more elaborate numerical figures such as magic circles and magic
hexagons. A variant of the idea of a magic square is that of a Latin square, an ç ÷ ç
array in which each of ç letters appears once in each row and once in each column.
It is easy to construct such a square by writing the letters down in order along the
first row and then cyclically permuting them by one step in each subsequent row.
To make the problem harder, mathematicians beginning with Euler in 1781 have
sought pairs of orthogonal Latin squares: that is, two Latin squares that can be
superimposed in such a way that each of the n^2 possible ordered pairs of letters
occurs exactly once. An example, given by Biggs (1979, p. 123), with one of the
squares using Greek letters for additional clarity, is
Modern combinatorics. The usefulness of combinatorics in elementary probability
has already been noted. It is an interesting exercise to compute, for example, the
probability of a particular poker hand, say a full house, and see why the rules of
the game make three of a kind a better hand (because less likely) than two pairs.
The strongest impetus to combinatorial studies in Europe, however, came from
Gottfried Wilhelm Leibniz (1646-1716), who is best remembered for his brilliant
discoveries in the calculus. He was also a profound philosopher and a diplomat
with a deep interest in Oriental cultures. Many fundamental results are found in
his De arte combinatoria, published in 1666. In this work Leibniz gave tables of
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