Editing Latin square (section)

==Applications==

===Statistics and mathematics===
*In the [[design of experiments]], Latin squares are a special case of ''row-column designs'' for two [[blocking (statistics)|blocking factors]].<ref>{{citation |author-link=Rosemary A. Bailey|first=R.A.|last=Bailey|chapter=6 Row-Column designs and 9 More about Latin squares|title=Design of Comparative Experiments|publisher=Cambridge University Press|year=2008 |isbn=978-0-521-68357-9|chapter-url=http://www.maths.qmul.ac.uk/~rab/DOEbook|mr=2422352}}</ref><ref>{{citation |last1=Shah|first1= Kirti R.|last2=Sinha|first2= Bikas K.| chapter=4 Row-Column Designs|title=Theory of Optimal Designs |series=Lecture Notes in Statistics| volume=54 | publisher=Springer-Verlag | year=1989 | pages=66–84 |isbn=0-387-96991-8 |mr=1016151}}</ref>
*In [[algebra]], Latin squares are related to generalizations of [[group theory|groups]]; in particular, Latin squares are characterized as being the [[multiplication table]]s ([[Cayley table]]s) of [[quasigroup]]s. A binary operation whose table of values forms a Latin square is said to obey the [[Quasigroup#Algebra|Latin square property]].

===Error correcting codes===
Sets of Latin squares that are [[Graeco-Latin square|orthogonal]] to each other have found an application as [[error correcting codes]] in situations where communication is disturbed by more types of noise than simple [[white noise]], such as when attempting to transmit broadband Internet over powerlines.<ref name="CKL">{{cite journal | last1 = Colbourn | first1 = C.J. | author-link = Charles Colbourn | last2 = Kløve | first2 = T. | last3 = Ling | first3 = A.C.H. | year = 2004 | title = Permutation arrays for powerline communication | journal = IEEE Trans. Inf. Theory | volume = 50 | pages = 1289–1291 | doi=10.1109/tit.2004.828150| s2cid = 15920471 }}</ref><ref name="NS">''Euler's revolution'', New Scientist, 24 March 2007, pp 48–51</ref><ref name="SH">{{cite journal | last1 = Huczynska | first1 = Sophie | year = 2006| title = Powerline communication and the 36 officers problem | journal = Philosophical Transactions of the Royal Society A | volume = 364 | issue = 1849| pages = 3199–3214 | doi=10.1098/rsta.2006.1885| pmid = 17090455 | bibcode = 2006RSPTA.364.3199H | s2cid = 17662664 }}</ref>

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.

<div class="center"><math display="block">
\begin{matrix}
A\\
B\\
C\\
D\\
\end{matrix}

\begin{bmatrix}
 1 & 2 & 3 & 4 \\
 2 & 1 & 4 & 3 \\
 3 & 4 & 1 & 2 \\
 4 & 3 & 2 & 1 \\
 \end{bmatrix}
\quad

\begin{matrix}
E\\
F\\
G\\
H\\
\end{matrix}

\begin{bmatrix}
1 & 3 & 4 & 2\\
2 & 4 & 3 & 1\\
3 & 1 & 2 & 4\\
4 & 2 & 1 & 3\\
\end{bmatrix}
\quad
\begin{matrix}
I\\
J\\
K\\
L\\
\end{matrix}

\begin{bmatrix}
1 & 4 & 2 & 3\\
2 & 3 & 1 & 4\\
3 & 2 & 4 & 1\\
4 & 1 & 3 & 2\\
\end{bmatrix}
</math></div>

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:
<math display="block">\begin{matrix}12 & 12 & 123 & 124\end{matrix}</math>

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:
<math display="block">\begin{matrix}1 & 2 & 1234 & 4\end{matrix}</math>

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 proven 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.

===Mathematical puzzles===

[[File:Ramanujan_magic_square_construction.svg|thumb|Construction of [[magic_square#Extra_constraints|Ramanujan's birthday magic square]] from a 4&times;4 Latin square with distinct diagonals and day (D), month (M), century (C) and year (Y) values, and Ramanujan's birthday example]]
The problem of determining if a partially filled square can be completed to form a Latin square is [[NP-complete]].<ref>{{cite journal | author = C. Colbourn | author-link = Charles Colbourn | title = The complexity of completing partial latin squares | journal = Discrete Applied Mathematics | volume = 8 | pages = 25–30 | year = 1984 | doi = 10.1016/0166-218X(84)90075-1| doi-access = free }}</ref>

The popular [[Mathematics of Sudoku|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 nine particular 3×3 adjacent subsquares must also contain the digits 1–9 (in the standard version). See also [[Mathematics of Sudoku]].

The more recent [[KenKen]] and [[Strimko]] puzzles are also examples of Latin squares.

===Board games===

Latin squares have been used as the basis for several board games, notably the popular abstract strategy game [[Kamisado]].

===Agronomic research===

Latin squares are used in the design of agronomic research experiments to minimise experimental errors.<ref>[http://joas.agrif.bg.ac.rs/archive/article/59 The application of Latin square in agronomic research]</ref>