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Fearless Symmetry by Robert Gross, Avner Ash

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122 CHAPTER 10
Thus Z =
4
2
, and indeed you can check that x = 4, y = 2solvesthe
original system (10.1)–(10.2).
J ust by glancing at them, who would have thought that the num-
ber 19 was lurking in the system of equations (10.1)–(10.2)? But if
you change the constants on the right-hand side of the equations,
you are likely to find 19’s popping up in the denominators of the
solutions. This is related to the fact that the determinant of the
matrix A is 19 (see chapter 11).
There is a complete theory of this and it gives you very efficient
ways to solve many simultaneous equations in many unknowns, as
long as none of the exponents is greater than 1. If you have many
equations and unknowns, you should probably use a computer to
help you, but the computer will be using matrices. At the end of the
next chapter we will do a 3-by-3 example.
Digression: Graeco-Latin Squares
Matrices with entries that are not numbers can still have uses.
For example, a game from the 1950s consisted of a 6-by-6 grid of
squares and 36 pieces. The plastic pieces were of six different colors
and six different shapes, so that each piece was different from all
the others. The challenge was to arrange the pieces on the grid in
such a way that no row or column contained two pieces of the same
shape or same color. The manufacturer offered a prize of $1000 for
the first solution. Some people tried this for quite a while.
In fact, the eighteenth-century mathematician Leonhard Euler
had already publicized this problem, which has a long subsequent
history. Instead of plastic pieces, he worked with soldiers, each of
which had a rank and a regiment. It is customary nowadays to
think of a square matrix, say with n rows and n columns. The
set of entries consists of n
2
symbols of the form Aα,thatis,one
Roman letter chosen from a list of n Roman letters, and one Greek
letter chosen from a list of n Greek letters, in such a way that the
entries are all different from one another. (If n is greater than 24,
find lengthier alphabets.)
MATRICES 123
DEFINITION: A matrix of these n
2
entries satisfying the
condition that no row and no column contains the same Latin
or the same Greek letter twice is called a Graeco-Latin
Square.
For example,
Aα Bβ Cγ Dδ
Dγ Cδ Bα Aβ
Cβ Dα Aδ Bγ
Bδ Aγ Dβ Cα
is a 4-by-4 Graeco-Latin Square. It is easy to see that there is no
2-by-2 Graeco-Latin Square.
These squares are actually useful in experimental design, when
you want to test all possible combinations of two variables . By the
way, there is no 6-by-6 Graeco-Latin Square. Alas. (This fact was
proven by a very patient mathematician in 1901.)
In fact, after mathematicians showed that there is no 2-by-2
Graeco-Latin Square, and no 6-by-6 Graeco-Latin Square, it was
tempting to guess (as Euler did) that there was no 10-by-10 Graeco-
Latin Square. However, this turned out to be wrong (though you
are unlikely to find one just by using trial-and-error without a
computer).
3
3
There are 8-by-8 Graeco-Latin Squares. In fact, Euler knew that there were n-by-n
Graeco-Latin Squares for any odd n or n divisible by 4. Euler conjectured that all 4k + 2-
by-4k + 2 cases (for k 0) were impossible, but in 1959 it was proven that all of them
are possible, except for the 2-by-2 and 6-by-6 cases!

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