In the following $2$ part question I have answer part $1$ but I am a little confused how to use this to answer part $2$, looking for some help thanks!
Suppose that two orthogonal $4x4$ Latin Squares both have $1,2,3,4$ as the main diagonal. Is it possible for both of them to have the same $(2,3)$ entry?
My Solution:
the order along the main diagonal is also matched between the two squares, in which case the answer to the opening question is naturally "no", since paired entries $(1\;1),$ $(2\;2),$ $(3\;3),$ $(4\;4)$ already exist and another matching pair would thus break orthogonality.
Here is the question I am not sure how to answer
What does this answer tell you about the number of $4x4$ MOLS each of which has $1 2 3 4$ as the main diagonal?
Does this mean there is no $4x4$ MOLS with $1 2 3 4 $ as the main diagonal?
The $(2,3)$ entry of a Latin square with $1,2,3,4$ on the diagonal cannot be $2$ or $3$, so it can only be $1$ or $4$. This shows there are at most two mutually orthogonal Latin squares with $1,2,3,4$ on the diagonal. You can just fill in the two possibilities and complete the squares. $$1342\quad\quad 1423\\4213\quad\quad 3241\\2431\quad\quad 4132\\3124\quad\quad 2314$$ Now check that they are in fact orthogonal. I glanced and didn't see a problem. This shows that there are exactly two MOLS with $1,2,3,4$ on the diagonal.