Ladder puzzle 1/Solution
Red's main threat is the ladder starting at c7. If Red plays out this ladder, Blue can block it. So Red needs a helping stone somewhere on the right and on the second line from the bottom. R1 threatens the follow-up moves at *, which Blue must defend.
Red continues that way, using the stone at the right. Again, Blue has to defend.
Finally, Red plays a double threat. This stone is the ladder helper, and it also threatens to connect along the top.
Alternative Solution: f6
The stone at f6 is a ladder breaker for the ladder starting at c7, and it also threatens to connect via e5. This looks very strong, but Blue can still defend at e7, which is a ladder breaker and threatens a connection at the lower side.
Now Red plays f4 completing the win. f4 is connected to the lower-right group in two non-overlapping ways, through f5 and g4. f4 is also connected to the central group in two non-overlapping ways, through f3 and e5. Therefore, all of Red's pieces form a single group which is connected to both the top and bottom.