Markers can move forward one square, or can jump over a marker of either colour if there is an empty square to land on.
Markers are not removed from the board if jumped.
You DO NOT have to alternate moving black and white markers.
The x's move E or S and o's move N or W and the goal is to make the x's and o's change places in these two board positions and in the general case.
|x|x|x| W + E
Consider the middle row as empty. Every column can be solved as with
the one-dimensional "trading places" problem, and in the end the middle
row would be empty again.
So, a straightforward solution is: solve the middle row, and whenever
the empty space is in an unsolved column, solve the column before
continuing the middle-row-solution. As every square in the middle row
will become empty at some time during the solution, in the end all
columns and the middle row are solved.