Calculating Gröbner basis for Sudoku

I deal with groebner bases on regular basis in Mathematica, and 81 variables, and that many polynomials is most probably way too much, even for software that is better with dealing with groebner bases. It is quite hard to estimate the time and memory consumption for such calculations (monomial order also plays a huge role), but my initial reaction is that your problem is way too hard to solve on a regular desktop computer.