excerpt from pages 173-174:
The program will generate a Greco-Latin square in
which every Latin letter occurs once in each row and once in each
column, and each Greek letter (represented here by numbers instead)
occurs once for each row and once for each column. In addition,
each Latin letter and Greek letter (number) occur only once
together in the square. Thus, for a Greco-Latin square of 5x5 there
are 25 combinations of Latin letters and numbers which can be
placed in 25 positions. This means that there are 25! (about 10 to the 25th power)
permutations out of which some Greco-Latin squares must be found.
Kempthorne  says a Greco-Latin square of side 12 exists,
although he gives no solution or reference. He does say that Greco-Latin
squares of even numbered sides have not been enumerated
except as he shows for a side of 4 (and possibly for a side of 12).
It can be seen that in Latin squares of sides N = 3,4,5 or more the
number of 45° diagonals made from two or more quadrants number N+j
where j is 0,1,2,3..., so that squares of odd numbered sides have
an even j while even numbered sides have an odd j. The four-sided
(j = 1) square is a special case, while Greco-Latin squares with
even numbered sides of 6 or more (j is odd and 3 or more) do not
seem to exist.
The program first finds a general solution for a Greco-Latin
square with an odd number of letters per side of 3 or more. For
example for a 5x5 square, this is done by beginning the first row
with ABCDE. The last letter of the row, E, starts on the next row
down and arrangement of letters is subsequently ordered (EABCD).
This pattern is shifted for the third (DEABC) and all rows
resulting in a Latin square. The same procedure is done with
numbers but in reverse (right to left) so that the first row is
54321 and the second row is 43215. The two Latin squares of letters
and numbers are superimposed and result in a Greco-Latin square.
The algorithm does not work for squares with sides with an even
number of rows/columns.
By switching two of the columns one obtains another Greco-Latin
square. Similarly two rows can be switched to get two different
squares. This principle is then used to switch at random various
rows and columns to get many different Greco-Latin squares. It was
found that the number of random switches of rows or columns needed
to be at least as large as the number of rows/columns in order to
effect a significant randomization of the initial pattern. With a
4x4 square it is possible to make six different swaps of the rows
for a total of seven different squares. Also six (N=6) different
swaps of the columns allow a grand total of
1 + [SUM (j=1 to N)of(j-1)]² = 37
different Greco-Latin squares including the original
one. For a 5x5 there are 101 squares, for a 7x7 there are 442
squares. The general algorithm does not work for even numbered
sides, but a Greco-Latin solution exists for a side of four as
shown earlier . This unique solution is used with the row and
column randomization algorithm to obtain the 37 possible Greco-Latin
A Greco-Latin square could be used if one wanted to test the effects
of 7 pheromone blends each at 7 dosages in 7 areas of the forest
on each of 7 days. The Greco-Latin square would allow a more powerful
analysis since all treatments and dosages were tested at each
position and on each day, although each position or day did not
have identical treatments/dosages. Several texts discuss the
advantages and drawbacks of using Latin and Greco-Latin squares as
well as the statistical analysis of variance [1, 3, 4, 6, 8, 9].
1. V. L. Anderson and R. A. McLean, Design of Experiments. Marcel
Dekker, Inc., New York (1974).
3. W. G. Cochran and G. M. Cox, Experimental Designs. Wiley, New
4. B. E. Cooper, Statistics for Experimentalists. Pergamon Press,
6. M. N. Das and N. C. Giri. Design and Analysis of Experiments.
Wiley, New Delhi (1979).
8. W. T. Federer, Experimental Design. MacMillan Co., New York
9. O. Kempthorne, The Design and Analysis of Experiments. Wiley,
New York (1952).
10. J. A. Byers, Basic algorithms for random sampling and treatment
randomization. Comput. Biol. Med. 21, 69 (1991).
Other related references:
Byers, J.A. 1991. BASIC algorithms for random sampling and treatment
randomization. Computers in Biology and Medicine 21:69-77.
Byers, J.A. 1996. Random selection algorithms for spatial and
temporal sampling. Computers in Biology and Medicine