There are too many different cases to consider, use a brute-force approach to get AC on the first attempt with one page of code:

1) write a function that checks that the situation is good;

2) for every coordinate make a list of four interesting values (all values taken by existing potatoes and some others if this is not enough);

3) try all possible ways to move none, one, or two potatoes, so that the coordinates of the new position are from the interesting values.

Disagree.
There're only 7 cases to consider.