FINITE GEOMETRIES

244 12 LATIN SQUARES

There are n2 ordered pairs (a, b) corresponding to the n2 points of the plane. These points also correspond to the n2 cells of an n × n square where (a, b) refers to the cell in the ath row and bth column. We fill these cells with numbers in n − 1 different ways to produce n − 1 mutually orthogonal latin squares of order n.

Consider any complete set of parallel lines that are not parallel to either axis. Label the n parallel lines 0, 1, 2, . . . , n − 1 in any manner. Through each point, there is exactly one of these lines. In the cell (a, b) place the number of the unique line on which the point (a, b) is found. The numbers in these cells form a latin square of order n on {0, 1, . . . , n − 1}. No two numbers in the same row can be the same, because there is only one line through two points in the same row, namely, the line parallel to the x-axis. Hence each number appears exactly once in each row and, similarly, once in each column.

There are n − 1 sets of parallelism classes that are not parallel to the axes; each of these gives rise to a latin square. These n − 1 squares are mutually orthogonal because each line of one parallel system meets all n of the lines of any other system. Hence, when two squares are superimposed, each number of one square occurs once with each number of the second square.

Conversely, suppose that there exists a set of n − 1 mutually orthogonal latin squares of order n. We can relabel the elements, if necessary, so that these squares are based on S = {0, 1, 2, . . . , n − 1}. We define an affine plane with S2 as the set of points. A set of n points is said to lie on a line if there is a latin square with the same number in each of the n cells corresponding to these points, or if the n points all have one coordinate the same. It is straightforward to check that this is an affine plane of order n.

Corollary 12.9. There exists an affine plane of order n whenever n is the power of a prime.

Proof. This follows from Theorem 12.5.

The only known affine planes have prime power order. Because of the impossibility of solving Euler’s officer problem, there are no orthogonal latin squares of order 6, and hence there is no affine plane of order 6. In 1988, by means of a massive computer search, Lam, Thiel, and Swiercz showed that there was no affine plane of order 10. See Lam [39] for the story behind this two-decade search. By Theorem 12.8, there cannot exist nine mutually orthogonal latin squares of order 10. However, two mutually orthogonal latin squares of order 10 have been found, but not three such squares. Computers have also been used to search for many sets of mutually orthogonal latin squares of low order. See Chapter 2 of Laywine and Mullen [40] for further results on mutually orthogonal latin squares.

By the method of Theorem 12.8, we can construct an affine plane of order n from the Galois field GF(n) whenever n is a prime power. The set of points is

P = GF(n)2 = {(x, y)|x, y GF(n)}.

Figure 12.3. Affine plane of order 4 with the points of the line y = αx + α2 enlarged.

It follows from Proposition 12.4 that a line consists of points satisfying a linear equation in x and y with coefficients in GF(n). The slope of a line is defined in the usual way and is an element of GF(n) or is infinite. Two lines are parallel if and only if they have the same slope.

For example, if GF(4) = Z2(α) = {0, 1, α, α2}, the 16 points of the affine plane of order 4 are shown in Figure 12.3. The horizontal lines are of the form

y = constant

and have slope 0, whereas the vertical lines are of the form x = constant

and have infinite slope. The line

y = αx + α2

has slope α and contains the points (0, α2), (1, 1), (α, 0) and (α2, α). This line is parallel to the lines y = αx, y = αx + 1, and y = αx + α.

Given an affine plane of order n, it is possible to construct a projective plane of order n by adding a “line at infinity” containing n + 1 points corresponding to each parallelism class, so that parallel lines intersect on the line at infinity. The projective plane of order n has n2 + n + 1 points and n2 + n + 1 lines. Furthermore, any projective plane gives rise to an affine plane by taking one line to be the line at infinity. Hence the existence of a projective plane of order n is equivalent to the existence of an affine plane of the same order.

MAGIC SQUARES

Magic squares have been known for thousands of years, and in times when particular numbers were associated with mystical ideas, it was natural that a

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Figure 12.4. “Melancholia,” an engraving by Albrecht Durer¨. In the upper right there is a magic square of order 4 with the date of the engraving, 1514, in the middle of the bottom row. [Courtesy of Staatliche Museen Kupferstichkabinett, photo by Walter Steinkopf.]

square that displays such symmetry should have been deemed to have magical properties. Figure 12.4 illustrates an engraving by Durer,¨ made in 1514, that contains a magic square. Magic squares have no applications, and this section is included for amusement only.

A magic square of order n consists of the integers 1 to n2 arranged in an n × n square array so that the row sums, column sums, and corner diagonal sums are all the same.

The sum of each row must be n(n2 + 1)/2, which is 1/n times the sum of all the integers from 1 to n2. For example, in Durer’s¨ magic square of Figure 12.4, the sum of each row, column, and diagonal is 34.

It is an interesting exercise to try to construct such squares. We show how to construct some magic squares from certain pairs of orthogonal latin squares. See Ball et al. [38] and Laywine and Mullen [40] for other methods of constructing magic squares.

Let K = (kij ) and L = (lij ) be two orthogonal latin squares of order n on the set S = {0, 1, . . . , n − 1}. Superimpose these two squares to form a square M = (mij ) in which the elements of M are numbers in the base n, whose first digit is taken from K and whose second digit is taken from L. That is, mij = n · kij + lij . Since K and L are orthogonal, all possible combinations of two elements from S occur exactly once in M. In other words, all the numbers from 0 to n2 − 1 occur in M.

Now add 1 to every element of M to obtain the square M = mij , where

mij = mij + 1.

Lemma 12.10. The square M contains all the numbers between 1 and n2 and is row and column magic; that is, the sums of each row and of each column are the same.

Proof. In any row or column of M, each number from 0 to n − 1 occurs exactly once as the first digit and exactly once as the second digit. Hence the sum is

(0 + 1 + · · · + n − 1)n + (0 + 1 + · · · + n − 1)

= (n + 1)(n − 1)n/2 = n(n2 − 1)/2.

Therefore, each row or column sum of M is n(n2 − 1)/2 + n = n(n2 + 1)/2.

Example 12.11. Construct the square M from the two orthogonal latin squares, K and L, in Table 12.13.

Solution. Table 12.13 illustrates the superimposed square M in base 3 and in

n(n − 1)/2, then the square M derived from K and L is a magic square of order n.