The applet below offers you two problems: one simple and one less simple. In the simple one,
you are requested to arrange numbers in a square matrix so as to have every number just once in
every row and every column. The second problem imposes one additional condition:
the arrangement must be symmetric with respect to the main diagonal (the one from the NorthWest to the SouthEast corner.)
A latin square of order n is an n by n array of n symbols in which
every symbol occurs exactly once in each row and column of the
array. Here are two examples.
Latin square of order 2  Latin square of order 3 
a b b a  x y z z x y y z x 
The great mathematician Leonhard Euler introduced latin squares in
1783 as a "nouveau espece de carres magiques", a new kind of magic
squares. He also defined what he meant by orthogonal latin
squares, which led to a famous conjecture of his that went unsolved
for over 100 years. In 1900, G. Tarry proved a particular case of the
conjecture. It was shown in 1960 by Bose, Shrikhande, and Parker that,
except for this one case, the conjecture was false.
You can get a bunch more latin squares (but only one more of the 2
by 2) by permuting rows, columns, and/or symbols in any
combination. For example, the latin squares below are derived from
the 3 by 3 latin square above.
Interchange last two rows 
Interchange last two rows and first and last columns 
Interchange symbols y and z 
x y z y z x z x y  z y x x z y y x z  x z y y x z z y x 

In one sense all of these latin squares of order 3 are all the same.
The original latin square could be a table of information and the
other tables just personal taste on how to display the information
and code the data. For example, suppose a dating service wants to
schedule dates for Ann, Bea, and Carol with Al, Brian, and Carl on
Monday, Tuesday, and Wednesday in such a way that each woman dates
each man in the three days. The dating service might display the
schedule as follows, with x, y, and z codes for Monday, Tuesday,
and Wednesday.
 Al  Brian  Carl 
Ann  x  y  z 
Bea  z  x  y 
Carol  y  z  x 
It is a matter of taste how the dating service chooses to order the
people and code the days, so the other three latin squares above
could display exactly the same information.
Let's say two latin squares are isomorphic iff one can be
transformed into the other by a combination of permuting rows,
columns, and symbols. Isomorphic comes from greek; iso meaning
same and morph meaning form.
Consider the three latin squares constructed below. The latin
square A of order 4 below is constructed by cyclically permuting the
symbols in the first row for subsequent rows.
1 2 3 4
2 3 4 1
3 4 1 2
4 1 2 3
The latin square B arises as the multiplication table for the
nonzero integers modulo 5:
1 2 3 4
2 4 1 3
3 1 4 2
4 3 2 1
Computers do modulo 2 arithmetic on strings of zeros and ones.
Here is the addition table for two bit strings:
00 01 10 11
01 00 11 10
10 11 00 01
11 10 01 00

changed to symbols 1 to 4 
1 2 3 4
2 1 4 3
3 4 1 2
4 3 2 1

The rightmost array above is the latin square C.
Latin square B can be obtained from A by interchanging the symbols
3 and 4 followed by interchanging the last two rows and last two
columns. However, no permuting of rows, columns, and/or symbols
will produce latin square C from A. Why is that? Taking a cue
from our leadin to the definition of isomorphism, note that latin
square C has six 2 by 2 subtables which are latin squares
involving the symbol 1. But in latin square B there is no symbol
which is part of six 2 by 2 subtables which are latin squares.
Since a combination of permuting rows, columns, and symbols
preserves the number of 2 by 2 subtables which are latin squares
containing a common symbol, latin squares B and C are not
isomorphic.