Above is a Graeco Latin Square of order four. It has all the sixteen double digited entries possible with the digits "1", "2", "3" and "4". On top of it, the first digits form a Latin Square having no duplicate in any row or in any column and the second digits also form a Latin Square. It is a Statistical Designing tool invented in 1725.
Now sixteen entries can be arranged in a four by four array in 16! ≈ 1013 ways and only 10,368 of them are Graeco Latin Squares. So it is a hard task to search out an order four Graeco Latin Square. There are various methods to construct a GLS and a 1959 paper by Bose, Srikhande and Parker confirms GLS is constructable for all orders except two and six.
Starting from a random cell with a random entry one can use two interlocking transversals with the starting entry as the common point to construct the order four GLS. Below I demo an example.
Let me start with a random entry "21" at the (2,2) cell.
Above is a Left Transversal of 2 where the entries follow the rules of a Graeco Latin Square and have the first digits as "2".
I have added three more entries following the rules of Graeco Latin Square all having "1" as the second digit. This is a Right Transversal of "1".
Now the GLS is uniquely constructable by unique choices. I use a ppt search macro to derive the GLS as below.
A green cell has the entry that is unique to the cell. A blue cell has the entry for which the cell is unique. Here I publish the construction the proof which is still alluring me.
One can think of a minimal number of entries that defines a GLS uniquely. Such a set is known as a Critical Set. We believe a Critical Set of an order four GLS has four entries. Whereas it is impossible to create a GLS Order Four uniquely with only three entries. I have an algorithm to create a Critical Set of order four GLS but that does not cover all the Critical Sets of an order four GLS.
The concept of Critical Sets could be extended to Latin Squares too. Below is a study on the minimum sizes achieved so far of the Critical Sets for both Latin Squares and Graeco Latin squares.
QUESTION I: How do one prove that the above construction is logically correct?
QUESTION II: How many entries are there in a Critical Set of GLS Order three?
REFERENCE: Wikipedia, My Previous Blog