I have to make this rule guess:
White has a move which is a successful ladder capture or ladder escape.
Taking “ladder” in the sense of a one-liberty-path.
- Green koan 249: White can obviously escape the ladder;
- Green koan 250: White can capture a Black stone by playing at tengen, and this is a “ladder” capture in the sense that it follows a one-liberty-path.
- Red koan 251 and 252: all stones have too many liberties, there are no ladder move.
- Red koan 254: Black has a successful ladder capturing move, capturing the White stone on the second line; but White doesn’t have any successful ladder capture nor ladder escape.