9-Rook problem in 3D / weak version of sudoku So here I have a 9*9 grid, and I have to fill each row and each column with 1~9, but without the constraint about 3*3 boxes as in traditional sudoku. Suppose the grid is blank at first, in how many different ways can I fill the boxes? Is there a (simple) mathematical way to compute this or do I have to write the DLX thingy? And is there a ready answer for this? Taking/not taking symmetry into account are both fine.

Without the 3x3 box constraints, this is just counting Latin squares (Wikipedia, Mathworld).

The number of Latin squares of size $n\times n$ is sequence A002860 in OEIS. See also A000315.

The short answer is that there is no known formula, but the growth is so fast that you won't get far trying to count them by brute force however good your heuristic.