Ladder puzzle 1/Solution
Solution 1: e5 then c7
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. Red 1.e5 threatens the follow-up moves at *, which Blue must defend.
The potential connection at d7 passes through e6, c7, c8, and d8. The potential connection at f6 passes through e6, f5, and Template IIIa. These two potential connections only overlap at the two hexes marked +, so Blue must play there. However, if Blue plays 2.d8, Red wins with the following sequence of forcing moves:
Therefore, Blue must play 2.e6. Red can now play the ladder at c7, break the ladder at g7, and win:
Solution 2: e5 then f5
Alternatively, after 1.e5 2.e6, Red could have continued the 4th row ladder with 3.f5, to which Blue may reply 4.f6. This allows Red to play a double threat 5.g7. This stone is the ladder helper, and it also threatens to connect along the top. Either way, Red wins.
However, this play is more complicated to analyze, because Blue may also respond in a number of other places instead of f6. If Blue plays 4.e8, then 5.g7 still works, albeit for slightly different reasons:
Note that in the final position, the red stone at e7 is connected up via * and down via +, so it is a winning position.
If Blue plays 4.f7 or anything to the right of it, Red gets their ladder escape at 5.e7:
Again, e7 is connected upwards via *.
Solution 3: f6
The stone at f6 (essentially Tom's move, but requiring a bit less space than usual due to h4) is a ladder escape for the ladder starting at c7, and it also threatens to connect via e5. Blue has no choice but to defend at e7. Red continues like this:
Now Red plays f4 completing the win. f4 is connected to the bottom edge via a trapezoid, and to Red's central group via double threat at f3 and e5. Therefore, all of Red's pieces form a single group which is connected to both the top and bottom.
Solution 4: switchback
Red can also use h4 to do a switchback. To set this up, Red starts playing the 2nd row ladder and breaks the ladder at f7:
Now the killer move is f5, which connects to Red's main group by double threat, and is also connected to h4. Blue is forced to play at f6, but Red reconnects with h5 via edge template IV-2-d.
This method works no matter the size of the board or how far away the 5th row stone is. If h5 is further to the right, the 2nd row ladder will just turn into a 4th row ladder going the opposite way, eventually connecting with d5.