Quadrangle criterion for a matrix I need help understanding the quadrangle criterion. First of all, I find it very hard to find anything related to it. The only two things I came up with are these:


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*"Reconstruction of Multiplication Tables" by P. Vojtechovsky and

*"On Minimum Distances and the Quadrangle Criterion" by Ales Drapal
And of course I found this question here:
Latin squares that don't come from group tables?
So I think I got a feeling for what the quadrangle criterion means. But as I cannot find any example for a matrix that satisfies this criterion, I am not sure if I understood it right.
What I think it means is this:
Say you have a $5\times 5$ matrix with the first two rows as follows:
$\array{1&2&3&4&5\cr3&4&5&1&2}$
The first thing that needs to hold is that every "subsquare", i.e. $\array{1&2\cr3&4}$ would be the first one in this example, has four distinct entries, as stated in the referenced Mathematics question, right? Thus, satisfied for the first and the second subsquare which is $\array{3&4\cr5&1}$.
So what I don't understand now is how they have to be equal to satisfy the criterion. In the referenced Mathematics answer, it is said that if you have two $2\times 2$ submatrices of the form
$$\array{a&b\cr c&d}$$
and
$$\array{a&b\cr c&x}$$
then $d=x$.
Does that hold for any $2\times 2$ submatrix in a symmetric matrix, even if you start with an even column (example: take the second and third column and first and second row: $\array{1&[2&3]&4&5\cr3&[4&5]&1&2}$)? Can somebody provide a matrix that satisfies the quadrangle criterion at best?
Thanks already in advance.
 A: First off, every subsquare does not have to have 4 distinct entries.  There is no condition like that.  There is also no need for symmetry.  The only basic assumption is that the matrix gives a Latin square.
The 2x2 subsquares don't have to be contiguous.  You can use the "subsquare" of the 4 intersections of rows 1 and 2 and columns 1 and 9, for instance.  We call them "quadrangles" to avoid confusion with contiguous subsquares.
Voytchekovsky states the criterion nicely.  We are comparing two different quadrangles.  Suppose the entries in three corners of the quadrangles match.  The quadrangle condition is that the fourth corner entries should also match.
In your example, you take any two rows and any two columns, say rows 1 and 2 and columns 3 and 5: the pattern is
3 5 / 5 2.
Now suppose you take rows 1 and 5 and columns 4 and 2 (in that order, just to show order is not required) and you find entries
3 5 / 5 x.
If x = 2, the quadrangle criterion is satisfied for these two quadrangles.  If x is not 2, the matrix fails the quadrangle criterion.
