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.