A Latin square is said to be reduced (also, normalized or in standard form) if its first row and first column are in natural order. For example, the Latin square above is reduced because both its first row and its first column are 1,2,3 (rather than 3,1,2 or any other order). We can make any Latin square reduced by permuting (reordering) the rows and columns.
If each entry of an n × n Latin square is written as a triple (r,c,s), where r is the row, c is the column, and s is the symbol, we obtain a set of n2 triples called the orthogonal array representation of the square. For example, the orthogonal array representation of the first Latin square displayed above is
The orthogonal array representation shows that rows, columns and symbols play rather similar roles, as will be made clear below.
Many operations on a Latin square produce another Latin square (for example, turning it upside down).
If we permute the rows, permute the columns, and permute the names of the symbols of a Latin square, we obtain a new Latin square said to be isotopic to the first. Isotopism is an equivalence relation, so the set of all Latin squares is divided into subsets, called isotopy classes, such that two squares in the same class are isotopic and two squares in different classes are not isotopic.
Another type of operation is easiest to explain using the orthogonal array representation of the Latin square. If we systematically and consistently reorder the three items in each triple, another orthogonal array (and, thus, another Latin square) is obtained. For example, we can replace each triple (r,c,s) by (c,r,s) which corresponds to transposing the square (reflecting about its main diagonal), or we could replace each triple (r,c,s) by (c,s,r), which is a more complicated operation. Altogether there are 6 possibilities including "do nothing", giving us 6 Latin squares called the conjugates (also parastrophes) of the original square.
Finally, we can combine these two equivalence operations: two Latin squares are said to be paratopic, also main class isotopic, if one of them is isotopic to a conjugate of the other. This is again an equivalence relation, with the equivalence classes called main classes, species, or paratopy classes. Each main class contains up to 6 isotopy classes.
There is no known easily-computable formula for the number of n × n Latin squares with symbols 1,2,...,n. The most accurate upper and lower bounds known for large n are far apart. Here we will give all the known exact values. It can be seen that the numbers grow exceedingly quickly.
For each n, the number of Latin squares altogether is n! (n-1)! times the number of reduced Latin squares .
|n||reduced Latin squares of size n||all Latin squares of size n|
For each n, each isotopy class contains up to (n!)3 Latin squares (the exact number varies), while each main class contains either 1, 2, 3 or 6 isotopy classes.
|n||main classes||isotopy classes|
We give one example of a Latin square from each main class up to order 5.
They present, respectively, the multiplication tables of the following groups:
Firstly, the message is sent by using several frequencies, or channels, a common method that makes the signal less vulnerable to noise at any one specific frequency. A letter in the message to be sent is encoded by sending a series of signals at different frequencies at successive time intervals. In the example below, the letters A to L are encoded by sending signals at four different frequencies, in four time slots. The letter C for instance, is encoded by first sending at frequency 3, then 4, 1 and 2.
The encoding of the twelve letters are formed from three Latin squares that are orthogonal to each other. Now imagine that there's added noise in channels 1 and 2 during the whole transmission. The letter A would then be picked up as:
In other words, in the first slot we receive signals from both frequency 1 and frequency 2; while the third slot has signals from frequencies 1, 2 and 3. Because of the noise, we can no longer tell if the first two slots were 1,1 or 1,2 or 2,1 or 2,2. But the 1,2 case is the only one that yields a sequence matching a letter in the above table, the letter A. Similarly, we may imagine a burst of static over all frequencies in the third slot:
Again, we are able to infer from the table of encodings that it must have been the letter A being transmitted. The number of errors this code can spot is one less than the number of time slots. It has also been proved that if the number of frequencies is a prime or a power of a prime, the orthogonal Latin squares produce error detecting codes that are as efficient as possible.
The problem of determining if a partially filled square can be completed to form a Latin square is NP-complete.
The popular Sudoku puzzles are a special case of Latin squares; any solution to a Sudoku puzzle is a Latin square. Sudoku imposes the additional restriction that 3×3 subgroups must also contain the digits 1–9 (in the standard version).