Can one find three $10\times 10$ mutually orthogonal Latin squares?
Does anyone know if there is a mathematical "trick" in finding mutually orthogonal Latin squares? Or is it basically trial and error?
[Math] Find three $10\times10$ orthogonal Latin squares.
combinatoricslatin-squarematrices
Related Solutions
You’re asking about mutually orthogonal Latin squares. The answer to the specific question is that yes, it is possible to find three mutually orthogonal Latin squares of order $4$, and that is the maximum. You’ll find a little more information and a few references at OEIS A001438.
This may also be of interest. From the two 4x4 squares given in a previous answer, two 8x8 squares can be grown.
If we transpose the symbols thus;
The two squares are different only by the chosen symbols the cells contain.
a) Place the two latin squares side by side to give a rectangle.
b) Then swap rows 2 and 4 with the transposed rows 2 and 4 to give;
c) Then take the resulting rectangle from b) and copy it backwards below the rectangle resulting from b).
Then we get;
The same process may be performed again and again from the resulting squares, swapping each even numbered rows.
Best Answer
Although pairs of MOLS of order 10 are known ("Euler spoilers"), the existence of three mutually orthogonal latin squares of order 10 is an open problem.
You may find some details of computer-based searches for these in Erin Delise's M.Sc. thesis (2005). Note the first two bibliographic entries, for Bose, Chakravarti, and Knuth (1960-61).
There are "tricks" to reducing the general search space, but it remains of a formidable size despite its numerous symmetries. The first row of all three such matrices may be assumed to be $(1,2,3,\ldots,10)$, and the number of individual latin squares of order 10 with this fixed first row is 2750892211809148994633229926400. We may further restrict the first column of the first (of the three) latin squares to be $(1,2,3,\ldots,10)^T$, saving a factor of $9!$ in size of search space.
Once an intial pair of MOLS are fixed, it becomes tractable to make an exhaustive computer search for the third MOLS of order 10. Many attempts along this line have failed, although an "almost orthogonal" triple was reported by Franklin (1983); see Mohan, Lee, and Pokhrel (2006) for some references to the literature.
A set of 9 mutually orthogonal latin squares of order 10 would amount to the existence of a finite projective plane of order 10, but Lam (1991) reported the results of an extensive computer search proved this impossible.