Q:

Masyu-making game

Two players are playing a game on a $4\times4$ board.  The players take turns adding either a white circle or a black circle into any empty square, so that the board makes a solvable masyu puzzle (with one or more possible solutions).  The winner is the player who first makes the position uniquely solvable (i.e. a puzzle which only has one possible solution). 

What move should the first player start with? 

A:

Up to symmetries of the board, there aren't very many possible moves for the first player:

 
 There are only five moves: black in A, black in B, black in F, white in B, and white in F.

 Black in A doesn't work; the second player responds with black in K.
 Black in B doesn't work; the second player responds with black in O.
 Black in F doesn't work; the second player responds with black in P.
 White in B doesn't work; the second player responds with black in K.
 So the only possible answer is white in F.

Does this strategy work?   

 Let's check all possible responses to white in F (again, ignoring symmetry):
 Black in A makes the puzzle unsolvable. 
 Black in B: respond with black in L.
 Black in C makes the puzzle unsolvable. 
 Black in D: respond with black in J.
 Black in G: respond with black in M.
 Black in H: respond with black in M.
 Black in K makes the puzzle unsolvable. 
 Black in L: respond with black in B.
 Black in P: respond with white in B.

 White in A makes the puzzle unsolvable. 
 White in B: respond with black in P.
 White in C: respond with black in L.
 White in D makes the puzzle unsolvable. 
 White in G: respond with black in N.
 White in H: respond with black in J.
 White in K: respond with white in L.
 White in L: respond with white in K.
 White in P makes the puzzle unsolvable. 

 So this first move does indeed work, and lets the first player win on their next turn.