Latin Square is a mathematical tool used in designing statistical experiments in the areas of agriculture, industry etc. It fascinated mathematicians like Leonard Euler though. The concept of Latin Square or Graeco Latin Square started with the following arrangement of cards by  a French Mathematician Jacques Ozanam in 1725. The arrangement has all the 16 honors cards of a playing deck where no row and no column has duplicates in any suit or in any denomination. 

Such arrangement with six suits and six denominations was known as Thirty Six Officers Problem and Euler attempted it in late 1700. Euler failed in it and later in 1901 another French Mathematician Gaston Tarry proved its impossibility using exhaustive enumeration. Euler conjectured that Graeco Latin Squares do not exist for the order 4k+2 which was disproved by Prof. E T Parker of University of Illinois in 1959 while constructing a Graeco Latin Square or a Mutually Orthogonal Pair of Latin Square order 10. Later Prof R C Bose, Prof Srikhande and Prof Parker together proved that Graeco Latin Squares or a Pair of Mutually Orthogonal Latin Squares exist for all orders except 2 and 6. Probably this is true in higher dimensions too.

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. This is a problem Latin-square researchers discuss from time to time, but it's generally regarded as a very hard problem. It's one of those research problems which is best to avoid if you want to remain a functional mathematician. You may find some details of computer-based searches for these in Erin Delise's M.Sc. thesis (2005). 

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,…,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,…,10), saving a factor of 9! in size of search space.

Once an initial 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.

Below all three are Latin Squares of Order 10 where First and Second and First and Third are Mutually Orthogonal but Second and Third fail to be Mutually Orthogonal as they have 9 pairs of duplicates in the colored cells.

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. 

Practically, we don't believe in the existence of a triplet of order 10. Not at the moment. Theoretically, we haven't excluded the possibility that they exist. I'd guess they do exist, but I wouldn't be surprised if they don't.

QUESTION I : How easy is to construct a Graeco Latin Square of odd order? 

QUESTION II : Could you construct a triplet of order 9?

REFERENCE : Find three  10×10 orthogonal Latin squares