Playing with Latin SquaresAuthor(s): Michael CorneliusSource: Mathematics in School, Vol. 21, No. 5 (Nov., 1992), pp. 28-29Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30214931 .

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Sq by Michael Cornelius, University of Durham

A LATIN SQUARE of order n consists of the numbers 0, 1, 2, 3, ... (n - 1) arranged in a square n x n grid so that in every row and every column each of the numbers appears once and only once. Figure 1 gives examples of 3 x 3 and 5 x 5 Latin Squares.

0 1 2

2 0 1

1 2 0

0 4 3 2 1

1 0 4 3 2

2 1 0 4 3

3 2 1 0 4

4 3 2 1 0 Fig. 1

Latin Squares can provide useful material for the teach- ing of mathematics at almost any level and within almost any section of the National Curriculum. They introduce mathematical patterns, involve facility with number, have real life applications and can be the vehicle for some interesting historical background.

The following are ideas which could be pursued in the classroom.

How many 3 x 3 Latin Squares can be found? By investigation, using trial and error or through a system- atic approach, the following 12 squares can be produced (Figure 2).

Fig. 2 0 1 1 0

1 0 0 1

1 0 1 0

1 0 I B I I0 1

2 0 0 2

o 2 2 20..2

2 0 0 2

1 2 2 1

2 1 1 2

1 2 2 1

1 2 2 1

2 1 1 2

1 2 2 1

These 12 squares can stimulate discussion on symmetry, reflection and rotation. It is easy to see that they are all really examples of the single pattern in Figure 3.

Fig. 3

The game of "Latino" This is a game of deduction in the style of "Mastermind". Two players each make up a Latin Square of agreed size (say 6 x 6). The squares are hidden and, asking questions in turn, each player seeks to deduce the opponent's square. A question can ask for the sum of any 2/3 adjacent numbers e.g., in Figure 4:

if you ask "D345" the answer is "12" if you ask "DE4" the answer is "7".

28 Mathematics in School, November 1992

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Fig. 4 A B C D E F

1 2 3 5 1 0 4

2 3 5 4 2 1 0

3 4 0 3 5 2 1

4 0 1 2 3 4 5

5 1 2 0 4 5 3

6 5 4 1 0 3 2

When one player thinks they know the opponent's square, the game ends. Both players produce copies of what they think the squares look like and the one who has most numbers correct is the winner. It is probably best to start with a fairly small size Latin Square but games with 8 x 8 or even 10 x 10 squares can be great fun.

Latin Squares in "Real Life" Suppose that you wish to try out 4 different brands of tyre on a car. There may be reasons, due to loading etc, why front/back or nearside/offside are subject to particular wear. One way of testing would be to put one each of the four different brands on the car and to rotate the types weekly before checking for wear. A Latin Square arrangement will give the plan for the tyres (Figure 5).

Fig. 5

Front Front Back Back Left Right Left Right

Week 1 Brand A Brand B Brand C Brand D Week 2 Brand D Brand A Brand B Brand C Week 3 Brand C Brand D Brand A Brand B Week 4 Brand B Brand C Brand D Brand A

A farmer may wish to try out (say) four kinds of seed in a field. A Latin Square arrangement like the one above would provide a suitable plan for a trial with the different types of seed.

A Problem with Playing Cards Can the sixteen cards: Ace, King, Queen, Jack of Clubs, Diamonds, Hearts and Spades be arranged in a 4 x 4 grid so that each row and column contains exactly one of each denomination and exactly one of each suit?

A possible solution is given in Figure 6 which also illustrates how this can be seen as two Latin squares superimposed. Such squares are called "orthogonal".

For Latin Squares of prime order there is an elegant way of constructing orthogonal squares. For example, with a 5 x 5 square, if the rows and columns are labelled 0, 1, 2, 3, 4 then each entry can be thought of as a number pair (x, y). Starting with the grid in Figure 7 and working in arithmetic mod 5, if points are labelled:

Fig. 7 04 14 24 34 44

03 13 23 33 43

02 12 22 32 42

01 11 21 31 41

00 10 20 30 40

A ify=x (mod 5) B ify=x+ 1 (mod 5) C if y =x+2 (mod 5) D if y=x+3 (mod 5) E if y=x+4 (mod 5)

then we obtain a Latin Square. Following a similar process with the "lines" y = nx (n = 2, 3, 4) we obtain 3 more Latin Squares and any two of these will be orthogonal.

Euler's Problem Euler (1707-1783) suggested the following problem:

Is it possible to arrange 36 officers, 6 of each of 6 different ranks from 6 different regiments in a square so that each row and column contains exactly one of each rank and one of each regiment.

Such an arrangement is in fact impossible although it is fun to try to achieve a solution! Euler conjectured that it was impossible to produce such an array for squares of size n when n=4k+ 2 (k > 0). (It is clearly impossible for n = 2).

It was not until 1959 that it was proved that n= 2 and n= 6 are the only impossible cases. Thus it now becomes a challenge to complete a 10 x 10 square - imagine soldiers of 10 different ranks from 10 different countries and arrange them in a suitable square - it can be done!

References Ball, W. W. R. and Coxeter, H. S. M. (1974) Mathematical Recreations

and Essays, University of Toronto Press. Cornelius, M. and Parr, A. (1991) What's Your Game?, Cambridge

University Press. Fletcher, T. J. (ed) (1964) Some Lessons in Mathematics, Cambridge

University Press.

J Q K A S D C H JS QD KC AH

Q J A K H C D S QH JC AD KS

K A J Q D S H C KD AS JH QC

A K Q J C H S D AC KH QS JD

Fig. 6

Mathematics in School, November 1992 29

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