I'm trying to write a program that solves sudokus using a Gröbner basis. I introduced 81 variables $x_1$ to $x_{81}$, this is a linearisation of the sudoku board.
The space of valid sudokus is defined by:
for $i=1,\ldots,81$ : $F_i = (x_i - 1)(x_i - 2)\cdots(x_i - 9)$ This represents the fact that all squares have integer values between 1 and 9.
for all $x_i$ and $x_j$ which are not equal but in the same row, column or block: $G_{ij} = (F_i - F_j)/(x_i - x_j)$ This represents that the variables $x_i$ and $x_j$ can not be equal.
All these $F_i$ and $G_{ij}$ together define the space of valid sudokus. This consists of 891 polynomials.
Now to solve a sudoku we can add the clues to the space, so by example if the clue of a sudoku is the first square is a 5, then we add $(x_1 - 5)$ to the space. If we now take the Gröbner basis of this space we can directly see the solution for it.
I understand what I am doing this far. But I have trouble finding a computable manner for finding the Gröbner bases. I have successfully done everything for $4\times4$ sudokus (or so-called shidokus). But Maple nor Singular are giving me a result for the Gröbner basis of the $9\times9$ sudoku space. You can see the commands I gave to Maple here (First I define the 891 polynomials, then I ask for a basis of it) I read papers saying it's feasible although inefficient to do what I strive for but I don't see how to find the solution, as they don't include many implementation details. Can anyone point me to a direction, making this problem easier for Maple or other software?