Six die but eight live
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This applies to the life of any group which has no eyes but a solid row of connected stones on the second line. As long as the group is firmly enclosed on the second line, the number of external liberties is irrelevant.
With seven stones and gote, White ends up with a straight-three big eye. After 5, White is dead.
With six stones, even with sente, White ends up with a straight-three eye space. After 4, White is dead.
 It is assumed no surrounding conditions affect this local truth:
If there is a white stone at one of the points a, White can threaten to connect by descending to b and make a living shape next. Therefore, depending on the surrounding position, the points a and b serve as useful aji for White.
If Black ignores the threat, White escapes.
The following exercises include such, slightly varying, conditions.