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.
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.