Answer :
checkers, also called drafts, board game, one of the oldest games in the world. Checkers are played by two people facing each other on a chessboard with 64 light and dark squares, just like a chessboard.
with n=0 clearly 1 path.
with n=1 clearly 0 ways.
with n=2 clearly 0 ways.
with n=3 clearly 0 ways.
If n=4, then the columns must be 2,2,0 and the rows must be 2,2,0. So the tiles must form the corners of the rectangle, there are 3 ways to select the top and bottom sides of the rectangle and 3 ways to select the left and right sides of the rectangle. So a total of 9.
If n=5, consider how many per row, it can only be 3,2,0. And clearly this means that one column has exactly one chip, so zero paths.
If n=6, consider how many per row, it could be 3,3,0 or 2,2,2. If it is 3,3,0 then there are clearly 3 ways. If it is 2,2,2, there are two possible column arrangements: 3,3,0 or 2,2,2. The first produces 3 arrangements and the second 6 arrangements. 12 ways in total.
If n=7 then it doesn't just work that the two "missing" squares are in the same row/column, there are obviously 6×3=18 ways to pick the two missing tiles, so there are (97) −18=18 correct arrangements .
If n=8 each method clearly works, then 9.
If n=9 each method clearly works, then 1.
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