Latin Square Design

In general, a Latin square of order n is an n×n square such that each row (and each column) is a permutation (or an arrangement) of the same n distinct elements. Suppose you lead a team of four chess players to play four rounds of chess against another team of four players. If each player must play against a different player in each round, a possible schedule could be as follows:

Here, we assumed that the players are numbered 1 to 4 in each team. For instance, in round 1, player 1 of team A will play 1 of team B, player 2 of team A will play 3 of team B, and so on.

Suppose you like to test four types of fertilizers on tomatoes planned in your garden. To reduce the effect of soils and watering in your experiment, you might choose 16 tomatoes planned in a 4 × 4 square (i.e., four rows and four columns), such that each row and each column has exactly one plan that uses one type of the fertilizers. Let the fertilizers denoted by A, B, C, and D, then one possible experiment is denoted by the following square on the left.

This table tells us that the plant at row 1 and column 1 will use fertilizer A, the plant at row 1 and column 2 will use fertilizer C, and so on. If we rename A to 1, B to 2, C to 3, and D to 4, then we obtain a square in the (b) portion of the table, which is identical to the square in the chess schedule.

In mathematics, all three squares in the previous section (without the row and column names) are called a Latin square of order four. The name Latin square originates from mathematicians of the 19th century like Leonhard Euler, who used Latin characters as symbols.

Various Definitions

One convenience of using the same set of elements for both the row and column indices and the elements inside the square is that we might regard • as a function ∗ defined on the set {1,2,3,4}: ∗ (1,1) = 1, ∗ (1,2) = 3, and so on. Of course, we might write it as 1∗1 = 1 and 1∗2 = 3, and we can also show that (1∗3)∗4 = 4. This square provides a definition of ∗, and the square (b) in the previous section is called “the multiplication table” of∗.

In mathematics, if the multiplication table of a binary function, say ∗, is a Latin square, then that function, together with the set of the elements, is called quasigroup. In contrast, if ∗ is a binary function over the set

and satisfies the following properties, then the multiplication table ∗ is a Latin square of order n.

For all elements w, x, y, z eS,

The left-cancellation law states that no symbol appears in any column more than once, and the right-cancellation law states that no symbol appears in any row more than once. The unique-image property states that each cell of the square can hold at most one symbol.

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