A k x k Latin square experiment is one in which three experimental factors (one of which
is a primary experimental factor) have k levels each. It is an incomplete block experiment
with  k x k experimental combination with only one treatment run in each block (which is
constructed by the  two non primary experimental factors in the form of a square). In
these k x k experimental combination there is a full factorial structur in every pair of the
factor see Factor level combination table below. Any row (i.e., fixed level of a non
primary experimental factor) or column (i.e., fixed level of another non primary
experimental factor) has all the levels of the primary experimental variable.

                                                        Factor B



                 Factor A


              level          1            2           3
                 1          1            3          2
                 2          2            1          3
                 3          3            2          1

Green are the levels of Factor C.

Table. Factor level Combinations in a 3 x 3 Latin Square (see table above)

Factor A level Factor B level Factor C level
       1       1       1
       1       2       3
       1       3        2
       2       1       2
       2       2       1
       2       3        3
       3       1       3
       3       2       2
       3       3        1

Notice if the green column was missing then the first two have the 3 x 3 full factorial structure for
factors A and B.
The same is true if we dropped any of the three columns in the above table.