Number of possible states on a $6\times 6$ grid You have a 6x6 square grid. You have 18 black and 18 white pieces. How many ways can you fill the board (one piece per square) such that there are 3 black and 3 white pieces per column, and such that there are 3 black and 3 white pieces per row?
I've tried to do it purely combinatorally , just looking row by row, but it gets confusing very fast. Is there perhaps some way to do it utilizing symmetries, like how a solution translated horizontally or vertically is a solution, or how the 90 degree rotation around the center of a solution is a solution?

By considering the case of a $2 \times 2$ grid with $1$ black piece and $1$ white piece for each row and each column you can count $2$ ways (i.e. BW-WB and WB-BW). It is not too difficult to count patiently all cases for a $4 \times 4$ grid, totalling $90$ ways.
Searching OEIS with $2,90$, you can find OEIS sequence A058527 and, for the $6 \times 6$ grid the value $297200$.
