$(40$ pts$.)$ Mathematically formulate a general Sudoku puzzle for $99$ matrix. Refer to the

following text for discussion of the decision variable to formulate the mathematical program

for a Sudoku program.$[$Note that this formulation can have integer$/$binary variables.$]$

Decision Variables:

Each cell of a Sudoku puzzle can be expressed as $(i,j)$ where i corresponds to the row,

and $j$ corresponds to the column, and both i and $j$ vary between $1$ and $9.$ Each cell

$(i,j)$ can exhibit a state $k,$ where $1k9.$ The decision variable $x,$ should indicate

whether a cell $(i,j)$ takes a state $k,$ or not.

What would be the dimension $($and number of indices$)$ of decision variable $x?$ Is the

decision variable $x$ binary, integer or continuous?