There is a puzzle
Suppose a standard $8x8$ chessboard has two (arbitrary) squares removed. The only thing known is that these two squares have different colours. Is it always possible to place 31 dominoes of size 2x1 so as to cover all of these squares?
The answer is
and there is very clever and fast prove:
Consider a closed curve, which covers all cells on the chessboard. You can cover chess board by dominoes simply putting them along the curve.
If two cells with different colours are missing the curve breaks to two curves, each has even number of cells, therefore they always can be covered with dominos.
I would like to know is there a realistic brute-force solution.
There is of course a brute-force solution, which considers all 32*32 combinations and solves them. But of course a man can't do it. Therefore, please understand, what I asking: is there a brute-force solution, which reduces all possible cells placements to several (let's say up to 10) options and simply check them one by one? If yes - please, show me it. If no - please prove it.
If there is such an solution - how can one modify this puzzle to rid of it (that means to increase number of options one needs to consider to be bigger than 10)?