六角圍棋殘局B

Proof

Firstly, it's obvious that one tile must be played at a to stop White from immediately closing their larger group.

    

First Tile (position a)

The three possible tile rotations at position a are shown below. The first rotation loses immediately by closing the White group (left) while the second rotation delays defeat but sets up two winning threats for White (middle), both of which cannot be answered by Blue's second tile. The first tile must therefore be played at a in the third rotation (right); however, this sets up an immediate threat (dotted) which Blue must address with their second tile at either b or c.

    
Result: First tile must be third rotation at position a, second tile at b or c.

Second Tile (position b)

The three possible tile rotations at position b are shown below. The first two rotations set up immediate wins for White but the third rotation (right) looks more promising.

    

However, even the third rotation loses as White can set up two disjoint threats on their next move to win, as shown below. Blue must therefore play their second tile at position c.

    
Result: Second tile cannot be played at position b.

Second Tile (position c)

Given that position b is no good, Blue's second placement must be one of the rotations at position c shown below. The first rotation sets up an immediate win for White (left) while the second rotation sets up a line of three tips that also guarantees a win for White (middle). The only remaining choice for the second tile is therefore rotation three at position c (right).

    
Result: Second tile must be third rotation at position c.

This move offers Blue a glimmer of hope, however White then has at least two killer replies, one of which is shown below.

    

This puzzle demonstrates how even apparently simple positions can require considerable analysis. Blue has exactly 1 promising move out of a possible 126, and even the fact that it fails is not immediately obvious (only 2 White replies out of a possible 195 then win).