I have a $8\times 8$ matrix, where every element of the matrix is either $0$, $1$ or $2$, so I define $A$ as a $8\times 8$ ternary matrix.
The matrix needs to have these properties:
- In the rows $1,2$ and $3$ the number $1$ must appear exactly once and no number $2$ can appear (per row).
- In the rows $4,5$ and $6$ the number $2$ must appear exactly once and no number $1$ can appear.
- In the rows $7$ and $8$ the numbers $1$ and $2$ must appear exactly once per row.
- All properties of the rows are held in the columns too.
An example would be
\begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 1 \end{pmatrix}
How many matrices can we construct having these properties? What I find is that every configuration of an element is strongly connected to all the others, so I think the number cannot be very big.
